0,3

Sometimes also called lambda numbers.

Also denominators of continued fraction convergents to sqrt(2): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129.

Number of lattice paths from (0,0) to the line x=n-1 consisting of U=(1,1), D=(1,-1) and H=(2,0) steps (i.e., left factors of Grand Schroeder paths); for example, a(3)=5, counting the paths H, UD, UU, DU and DD. - Emeric Deutsch, Oct 27 2002

a(2*n) with b(2*n) := A001333(2*n), n >= 1, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = +1 (see Emerson reference). a(2*n+1) with b(2*n+1) := A001333(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = -1.

Bisection: a(2*n+1) = T(2*n+1, sqrt(2))/sqrt(2) = A001653(n), n >= 0 and a(2*n) = 2*S(n-1,6) = 2*A001109(n), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. - Wolfdieter Lang, Jan 10 2003

Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2, 7/5, 17/12, 41/29, ... converging to 2^(1/2). Sequence contains the denominators. - Amarnath Murthy, Mar 22 2003

This is also the Horadam sequence (0,1,1,2). Limit_{n->infinity} a(n)/a(n-1) = sqrt(2) + 1 = A014176. - Ross La Haye, Aug 18 2003

Number of 132-avoiding two-stack sortable permutations.

For n > 0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 3. - Herbert Kociemba, Jun 02 2004

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 2. - Herbert Kociemba, Jun 02 2004

Counts walks of length n from a vertex of a triangle to another vertex to which a loop has been added. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition of Pisot sequences. - David W. Wilson

Sums of antidiagonals of A038207 [Pascal's triangle squared]. - Ross La Haye, Oct 28 2004

The Pell primality test is "If N is an odd prime, then P(N)-Kronecker(2,N) is divisible by N". "Most" composite numbers fail this test, so it makes a useful pseudoprimality test. The odd composite numbers which are Pell pseudoprimes (i.e., that pass the above test) are in A099011. - Jack Brennen, Nov 13 2004

a(n) = sum of n-th row of triangle in A008288 = A094706(n) + A000079(n). - Reinhard Zumkeller, Dec 03 2004

Pell trapezoids (cf. A084158); for n > 0, A001109(n) = (a(n-1) + a(n+1))*a(n)/2; e.g., 1189 = (12+70)*29/2. - Charlie Marion, Apr 01 2006

(0!a(1), 1!a(2), 2!a(3), 3!a(4), ...) and (1,-2,-2,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 29 2007

Let C = (sqrt(2)+1) = 2.414213562..., then for n > 1, C^n = a(n)*(1/C) + a(n+1). Example: C^3 = 14.0710678... = 5*(0.414213562...) + 12. Let X = the 2 X 2 matrix [0, 1; 1, 2]; then X^n * [1, 0] = [a(n-1), a(n); a(n), a(n+1)]. a(n) = numerator of n-th convergent to (sqrt(2)-1) = 0.41421356... = [2, 2, 2, ...], the convergents being [1/2, 2/5, 5/12, ...]. - Gary W. Adamson, Dec 21 2007

A = sqrt(2) = 2/2 + 2/5 + 2/(5*29) + 2/(29*169) + 2/(169*985) + ...; B = ((5/2) - sqrt(2)) = 2/2 + 2/(2*12) + 2/(12*70) + 2/(70*408) + 2/(408*2378) + ...; A+B = 5/2. C = 1/2 = 2/(1*5) + 2/(2*12) + 2/(5*29) + 2/(12*70) + 2/(29*169) + ... - Gary W. Adamson, Mar 16 2008

Prime Pell numbers with an odd index gives the RMS value (A141812) of prime RMS numbers (A140480). - Ctibor O. Zizka, Aug 13 2008

From Clark Kimberling, Aug 27 2008: (Start)

Related convergents (numerator/denominator):

lower principal convergents: A002315/A001653

upper principal convergents: A001541/A001542

intermediate convergents: A052542/A001333

lower intermediate convergents: A005319/A001541

upper intermediate convergents: A075870/A002315

principal and intermediate convergents: A143607/A002965

lower principal and intermediate convergents: A143608/A079496

upper principal and intermediate convergents: A143609/A084068. (End)

Equals row sums of triangle A143808 starting with offset 1. - Gary W. Adamson, Sep 01 2008

Binomial transform of the sequence:= 0,1,0,2,0,4,0,8,0,16,..., powers of 2 alternating with zeros. - Philippe Deléham, Oct 28 2008

a(n) is also the sum of the n-th row of the triangle formed by starting with the top two rows of Pascal's triangle and then each next row has a 1 at both ends and the interior values are the sum of the three numbers in the triangle above that position. - Patrick Costello (pat.costello(AT)eku.edu), Dec 07 2008

Starting with offset 1 = eigensequence of triangle A135387 (an infinite lower triangular matrix with (2,2,2,...) in the main diagonal and (1,1,1,...) in the subdiagonal. - Gary W. Adamson, Dec 29 2008

Starting with offset 1 = row sums of triangle A153345. - Gary W. Adamson, Dec 24 2008

From Charlie Marion, Jan 07 2009: (Start)

In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:

let a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)

and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);

let b(k,0) = 1, b(k,1) = 2k+1; for n > 0, b(k,2n) = 2*b(k,2n-1) + b(k,2n-2)

and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).

For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.

In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then

k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and

b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);

for example, if k=1 and n=3, then a(1,n) = a(n+1) and

1*a(1,6)^2 - a(1,5)*a(1,7) = 1*169^2 - 70*408 = 1;

1*a(1,4)*a(1,6) - a(1,5)^2 = 1*29*169 - 70^2 = 1;

b(1,5)*b(1,7) - 1*b(1,6)^2 = 99*577 - 1*239^2 = 2;

b(1,5)^2 - 1*b(1,4)*b(1,6) = 99^2 - 1*41*239 = 2.

Cf. A001333, A142238, A142239, A153313, A153314, A153315, A153316, A153317, A153318.

(End)

Starting with offset 1 = row sums of triangle A155002, equivalent to the statement that the Fibonacci series convolved with the Pell series prefaced with a "1": (1, 1, 2, 5, 12, 29, ...) = (1, 2, 5, 12, 29, ...). - Gary W. Adamson, Jan 18 2009

It appears that P(p) == 8^((p-1/2)) mod p, p = prime; analogous to [Schroeder, p.90]: Fp == 5^((p-1)/2)) mod p. Example: Given P(11) = 5741, == 8^5 mod 11. Given P(17) = 11336689, == 8^8 mod 17 since 17 divides (8^8 - P(l7)). - Gary W. Adamson, Feb 21 2009

Equals eigensequence of triangle A154325. - Gary W. Adamson, Feb 12 2009

Another combinatorial interpretation of a(n-1) arises from a simple tiling scenario. Namely, a(n-1) gives the number of ways of tiling a 1 X n rectangle with indistinguishable 1 X 2 rectangles and 1 X 1 squares that come in two varieties, say, A and B. For example, with C representing the 1 X 2 rectangle, we obtain a(4)=12 from AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB, AC, BC, CA and CB. - Martin Griffiths, Apr 25 2009

a(n+1) = 2*a(n) + a(n-1), a(1)=1, a(2)=2 was used by Theon from Smyrna. - Sture Sjöstedt, May 29 2009

The n-th Pell number counts the perfect matchings of the edge-labeled graph C_2 x P_(n-1), or equivalently, the number of domino tilings of a 2 x (n-1) cylindrical grid. - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009

Number of units of a(n) belongs to a periodic sequence: 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1. - Mohamed Bouhamida, Sep 04 2009

As a fraction: 1/79 = 0.0126582278481... or 1/9799 = 0.000102051229...(1/119 and 1/10199 for sequence in reverse). - Mark Dols, May 18 2010

Limit_{n->infinity} (a(n)/a(n-1) - a(n-1)/a(n)) tends to 2.0. Example: a(7)/a(6) - a(6)/a(7) = 169/70 - 70/169 = 2.0000845... - Gary W. Adamson, Jul 16 2010

Numbers k such that 2*k^2 +- 1 is a square. - Vincenzo Librandi, Jul 18 2010

Starting (1, 2, 5, ...) = INVERTi transform of A006190: (1, 3, 10, 33, 109, ...). - Gary W. Adamson, Aug 06 2010

[u,v] = [a(n), a(n-1)] generates all Pythagorean triples [u^2-v^2, 2uv, u^2+v^2] whose legs differ by 1. - James R. Buddenhagen, Aug 14 2010

An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 21 and 336, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A078057. - Johannes W. Meijer, Aug 15 2010

Let the 2 X 2 square matrix A=[2, 1; 1, 0] then a(n) = the (1,1) element of A^(n-1). - Carmine Suriano, Jan 14 2011

Define a t-circle to be a first-quadrant circle tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the t-circle with radius 1. Then for n > 0, define C(n) to be the next larger t-circle which is tangent to C(n - 1). C(n) has radius A001333(2n) + a(2n)*sqrt(2) and each of the coordinates of its point of intersection with C(n + 1) is a(2n + 1) + (A001333(2n + 1)*sqrt(2))/2. See similar Comments for A001109 and A001653, Sep 14 2005. - Charlie Marion, Jan 18 2012

A001333 and A000129 give the diagonal numbers described by Theon from Smyrna. - Sture Sjöstedt, Oct 20 2012

Pell numbers could also be called "silver Fibonacci numbers", since, for n >= 1, F(n+1) = ceiling(phi*F(n)), if n is even and F(n+1) = floor(phi*F(n)), if n is odd, where phi is the golden ratio, while a(n+1) = ceiling(delta*a(n)), if n is even and a(n+1) = floor(delta*a(n)), if n is odd, where delta = delta_S = 1+sqrt(2) is the silver ratio. - Vladimir Shevelev, Feb 22 2013

a(n) is the number of compositions (ordered partitions) of n-1 into two sorts of 1's and one sort of 2's. Example: the a(3)=5 compositions of 3-1=2 are 1+1, 1+1', 1'+1, 1'+1', and 2. - Bob Selcoe, Jun 21 2013

Between every two consecutive squares of a 1 X n array there is a flap that can be folded over one of the two squares. Two flaps can be lowered over the same square in 2 ways, depending on which one is on top. The n-th Pell number counts the ways n-1 flaps can be lowered. For example, a sideway representation for the case n = 3 squares and 2 flaps is \\., .//, \./, ./_., ._\., where . is an empty square. - Jean M. Morales, Sep 18 2013

Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A005319(k)*(a(n-2k+1) - a(n-2k)) + a(n-4k) = A075870(k)*(a(n-2k+2) - a(n-2k+1)) - a(n-4k+2)). - Charlie Marion, Nov 26 2013

An alternative formulation of the combinatorial tiling interpretation listed above: Except for n=0, a(n-1) is the number of ways of partial tiling a 1 X n board with 1 X 1 squares and 1 X 2 dominoes. - Matthew Lehman, Dec 25 2013

Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A077444(k)*a(n-2k+1) + a(n-4k+2). This formula generalizes the formula used to define this sequence. - Charlie Marion, Jan 30 2014

a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 1, 1; 1, 1, 1; 0, 1, 1], [0, 1, 1; 0, 1, 1; 1, 1, 1], [0, 1, 0; 1, 1, 1; 1, 1, 1] or [0, 0, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

a(n+1) counts closed walks on K2 containing two loops on the other vertex. Equivalently the (1,1) entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1;1,2). - David Neil McGrath, Oct 28 2014

For n >= 1, a(n) equals the number of ternary words of length n-1 avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

This is a divisibility sequence (i.e., if n|m then a(n)|a(m)). - Tom Edgar, Jan 28 2015

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 03 2017

a(n) is the number of compositions (ordered partitions) of n-1 into two kinds of parts, n and n', when the order of the 1 does not matter, or equivalently, when the order of the 1' does not matter. Example: When the order of the 1 does not matter, the a(3)=5 compositions of 3-1=2 are 1+1, 1+1'=1+1, 1'+1', 2 and 2'. (Contrast with entry from Bob Selcoe dated Jun 21 2013.) - Gregory L. Simay, Sep 07 2017

Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal element for the weak ordering R. - J. Devillet, Sep 28 2017

Also the number of matchings in the (n-1)-centipede graph. - Eric W. Weisstein, Sep 30 2017

Let A(r,n) be the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only, and so A(r,0)=1. Let A_1(r,n) = Sum_{j=0..n} A(r,j) and let A_s(r,n) = Sum_{j=0..n} A_(s-1)(r,j). Then A_0(1,n) + A_2(3,n-4) + A_4(5,n-8) + ... + A_(2j) (2j+1, n-4j) = a(n) without the initial 0. - Gregory L. Simay, May 25 2018

(1, 2, 5, 12, 29, ...) is the fourth INVERT transform of (1, -2, 5, -12, 29, ...), as shown in A073133. - Gary W. Adamson, Jul 17 2019

Number of 2-compositions of n restricted to odd parts (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020

J. Austin and L. Schneider, Generalized Fibonacci sequences in Pythagorean triple preserving sequences, Fib. Q., 58:1 (2020), 340-350.

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

S.-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 941.

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 53.

John Derbyshire, Prime Obsession, Joseph Henry Press, 2004, see p. 16.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.1.

Shaun Giberson and Thomas J. Osler, Extending Theon's Ladder to Any Square Root, Problem 3858, Elementa, No. 4 1996.

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149.

Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

Manfred R. Schroeder, "Number Theory in Science and Communication", 5th ed., Springer-Verlag, 2009, p. 90.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 62.

N. J. A. Sloane and Simone Sandri, Table of n, a(n) for n = 0..1000 (First 500 terms from N. J. A. Sloane).

M. Abrate, S. Barbero, U. Cerruti, and N. Murru, Construction and composition of rooted trees via descent functions, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.

Paraskevas K. Alvanos and Konstantinos A. Draziotis, Integer Solutions of the Equation y^2 = Ax^4 + B, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.4.

Tewodros Amdeberhan, Solution to problem #10663 (AMM)

Ayoub B. Ayoub, Fibonacci-like sequences and Pell equations, The College Mathematics Journal, Vol. 38 (2007), pp. 49-53.

Ovidiu Bagdasar, Eve Hedderwick, and Ioan-Lucian Popa, On the ratios and geometric boundaries of complex Horadam sequences, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 63-70.

Aseem R. Baranwal and Jeffrey Shallit, Critical exponent of infinite balanced words via the Pell number system, arXiv:1902.00503 [cs.FL], 2019.

Elena Barcucci, Antonio Bernini, and Renzo Pinzani, A Gray code for a regular language, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.

M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7.

Latham Boyle and Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.

B. Bradie, Extensions and Refinements of some properties of sums involving Pell Numbers, Miss. J. Math. Sci 22 (1) (2010) 37-43.

Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, Combinatorial Interpretation of Generalized Pell Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.

Dorota Bród, On a New One Parameter Generalization of Pell Numbers, Annales Mathematicae Silesianae 33 (2019), 66-76.

Steve Butler, Jason Ekstrand, and Steven Osborne, Counting Tilings by Taking Walks in a Graph, A Project-Based Guide to Undergraduate Research in Mathematics, Birkhäuser, Cham (2020), see page 165.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Geoffrey B. Campbell and Aleksander Zujev, Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424 [math.NT], 2015.

Frédéric Chapoton, A note on gamma triangles and local gamma vectors, arXiv:1809.00575 [math.CO], 2018.

C. O. Chow, S. M. Ma, T. Mansour, and M. Shattuck, Counting permutations by cyclic peaks and valleys, Annales Mathematicae et Informaticae, (2014), Vol. 43, pp. 43-54.

M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.

Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.

C. M. da Fonseca, Unifying some Pell and Fibonacci identities, Applied Mathematics and Computation, Volume 236, Jun 01 2014, Pages 41-42.

Mahadi Ddamulira, On the x-coordinates of Pell equations which are products of two Pell numbers, arXiv:1906.06330 [math.NT], 2019.

E. Deutsch, A formula for the Pell numbers, Problem 10663, Amer. Math. Monthly 107 (No. 4, 2000), solutions pp. 370-371.

Antonio J. Di Scala, Nadir Murru, and Carlo Sanna, Lucas pseudoprimes and the Pell conic, arXiv:2001.00353 [math.NT], 2020.

E. S. Egge and T. Mansour, 132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers, arXiv:math/0205206 [math.CO], 2002.

Shalosh B. Ekhad and Tewodros Amdeberhan, Solution to Problem #10663.

C. Elsner, On Error Sums for Square Roots of Positive Integers with Applications to Lucas and Pell Numbers, J. Int. Seq. 17 (2014) # 14.4.4.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Ex. 1, pp. 237-238.

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.

Sergio Falcon, Generating Function of Some k-Fibonacci and k-Lucas Sequences, International Journal of Innovation in Science and Mathematics (2019) Vol. 7, Issue 2, 2347-9051.

Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.

Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.

Felix Flicker, Time quasilattices in dissipative dynamical systems, arXiv:1707.09371 [nlin.CD], 2017. Also SciPost Phys. 5, 001 (2018).

Robert Frontczak and Taras Goy, Mersenne-Horadam identities using generating functions, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 34-45.

Shaun Giberson and Thomas J. Osler, Extending Theon's Ladder to Any Square Root, College Mathematics Journal, May, 2004.

Juan B. Gil and Aaron Worley, Generalized metallic means, arXiv:1901.02619 [math.NT], 2019.

Taras Goy, Pell numbers identities from Toeplitz-Hessenberg determinants, Novi Sad J. Math., 49 (2) (2019), 87-94.

Martin Griffiths, Pell identities via a quadratic field, International Journal of Mathematical Education in Science and Technology, 2013.

R. P. Grimaldi, Tilings, Compositions, and Generalizations, J. Int. Seq. 13 (2010), 10.6.5.

M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.

Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

R. J. Hetherington, Letter to N. J. A. Sloane, Oct 26 1974

Gábor Hetyei, The type B permutohedron and the poset of intervals as a Tchebyshev transform, University of North Carolina-Charlotte (2019).

Nick Hobson, Python program for this sequence

Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.

A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.

A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252, 263.

Haruo Hosoya, What Can Mathematical Chemistry Contribute to the Development of Mathematics?, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 135

M. Janjić, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015), #15.4.7.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Tanya Khovanova, Recursive Sequences

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

C. J. Kirchen, Letter to N. J. A. Sloane, Feb 11 1974

Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16.

K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1.

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

Shirley Law, Hopf Algebra of Sashes, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 621-632.

H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 46.

Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

T. Mansour and M. Shattuck, A statistic on n-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 127-140.

A. Moghaddamfar and H. Tajbakhsh, More Determinant Representations for Sequences, Journal of Integer Sequences, 17 (2014), #14.5.6.

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170 [math.CO], 2018-2020.

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, (2019) [math].

Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.

Mariana Nagy, Simon R. Cowell, and Valeriu Beiu, Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight, arXiv:1902.05944 [math.HO], 2019.

Ahmet Öteleş, Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 109-120.

Ahmet Öteleş, Zekeriya Y. Karata, and Diyar O. Mustafa Zangana, Jacobsthal Numbers and Associated Hessenberg Matrices, J. Int. Seq., 21 (2018), #18.2.5.

Arzu Özkoç, Some algebraic identities on quadra Fibona-Pell integer sequence, Advances in Difference Equations, 2015, 2015:148.

Hao Pan, Arithmetic properties of q-Fibonacci numbers and q-Pell numbers, Discr. Math., 306 (2006), 2118-2127.

D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.

Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Raul Prisacariu, Generating k-Pell Infinite Series Using Whittaker's Formula, The Mathematics Enthusiast: Vol. 15 : No. 3 , Article 7, 2018.

C. Raissi and J. Pei, Towards Bounding Sequential Patterns, KDD'11, Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, 2011.

Franck Ramaharo, An approximate Jerusalem square whose side equals a Pell number, arXiv:1801.00466 [math.CO], 2018.

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.

John Riordan and N. J. A. Sloane, Correspondence, 1974

Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.

J. L. Schiffman, Exploring the Fibonacci sequence of order two with CAS technology, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012, Paper C027.

Jon E. Schoenfield, Prime factorization of a(n) for n = 1..300

James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.

Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers of odd index, Integers, 9 (2009), 53-64.

M. Shattuck, Combinatorial Proofs of Some Formulas for Triangular Tilings, Journal of Integer Sequences, 17 (2014), #14.5.5.

Nanci Smith, Problem B-82 An Integer Valued Function, Fib. Quart., 4 (1966), 374-375.

Yüksel Soykan, On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports (2019) Vol. 6, No. 1, Article No. AJARR.51635, 1-18.

Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article No. AIR.51824.

Yüksel Soykan, On generalized sixth-order Pell sequences, Journal of Scientific Perspectives (2020) Vol. 4, No. 1, 49-70.

Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.

Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.

Yüksel Soykan, Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Sum_{k=0..n} W_k^3 and Sum_{k=1..n} W_(-k)^3, Archives of Current Research International (2020) Vol. 20, Issue 2, 58-69.

Yüksel Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.

Yüksel Soykan, Mehmet Gümüş, and Melih Göcen, A Study On Dual Hyperbolic Generalized Pell Numbers, Zonguldak Bülent Ecevit University (Zonguldak, Turkey, 2019).

Robin James Spivey, Close encounters of the golden and silver ratios, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.

R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths

Ping Sun, Enumeration of standard Young tableaux of shifted strips with constant width, arXiv:1506.07256 [math.CO], 24 Jun 2015.

Tamas Szakacs, Convolution of second order linear recursive sequences I, Ann. Math. Inform. 46 (2016), 205-216.

Wipawee Tangjai, A Non-standard Ternary Representation of Integers, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.

Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).

A. Tekcan, M. Tayat, and M. E. Ozbek, The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.

Andrew Vince, The average size of a connected vertex set of a graph-explicit formulas and open problems, University of Florida (2019).

Ian Walker, Explorations in Recursion with John Pell and the Pell Sequence.

Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.

Eric Weisstein's World of Mathematics, Centipede Graph

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Pell Number

Eric Weisstein's World of Mathematics, Pell Polynomial

Eric Weisstein's World of Mathematics, Pythagoras's Constant

Eric Weisstein's World of Mathematics, Square Root

Eric Weisstein's World of Mathematics, Square Triangular Number

Meral Yasar and Durmus Bozkurt, Another proof of Pell identities by using the determinant of tridiagonal matrix, Appl. Math. Comput., 218 (2012), pp. 6067-6071.

Leon Zaporski and Felix Flicker, Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences, arXiv:1811.00331 [nlin.CD], 2018.

Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

Jianqiang Zhao, Finite Multiple zeta Values and Finite Euler Sums, arXiv preprint arXiv:1507.04917 [math.NT], 2015.

Index entries for "core" sequences

Index entries for sequences related to Chebyshev polynomials.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (2,1).

G.f.: x/(1 - 2*x - x^2). - Simon Plouffe in his 1992 dissertation.

G.f.: Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (2*k + x)/(1 + 2*k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 1 + k)/(1 + k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 3 - k)/(1 - k*x) ) may all be proved using telescoping series. - Peter Bala, Jan 04 2015

a(n) = 2*a(n-1) + a(n-2), a(0)=0, a(1)=1.

a(n) = ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/(2*sqrt(2)).

For initial values a(0) and a(1), a(n) = ((a(0)*sqrt(2)+a(1)-a(0))*(1+sqrt(2))^n + ((a(0)*sqrt(2)-a(1)+a(0))*(1-sqrt(2))^n)/(2*sqrt(2)). - Shahreer Al Hossain, Aug 18 2019

a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1. - Clark Kimberling

a(n) = Sum_{i, j, k >= 0: i+j+2k=n} (i+j+k)!/(i!*j!*k!).

a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).

a(2n) = 2*a(n)*A001333(n). - John McNamara, Oct 30 2002

a(n) = ((-i)^(n-1))*S(n-1, 2*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(-2, x)= -1.

Binomial transform of expansion of sinh(sqrt(2)x)/sqrt(2). E.g.f.: exp(x)sinh(sqrt(2)x)/sqrt(2). - Paul Barry, May 09 2003

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)2^k. - Paul Barry, May 13 2003

a(n-2) + a(n) = (1 + sqrt(2))^(n-1) + (1 - sqrt(2))^(n-1) = A002203(n-1). (A002203(n))^2 - 8(a(n))^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003

Unreduced g.f.: x(1+x)/(1 - x - 3x^2 - x^3); a(n) = a(n-1) + 3a(n-2) + a(n-2). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^(n-2k). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

Apart from initial terms, inverse binomial transform of A052955. - Paul Barry, May 23 2004

a(n)^2 + a(n+2k+1)^2 = A001653(k)*A001653(n+k); e.g., 5^2 + 70^2 = 5*985. - Charlie Marion Aug 03 2005

a(n+1) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^k/2. - Paul Barry, Aug 28 2005

a(n) = a(n-1) + A001333(n-1) = A001333(n) - a(n-1) = A001109(n)/A001333(n) = sqrt(A001110(n)/A001333(n)^2) = ceiling(sqrt(A001108(n)/2)). - Henry Bottomley, Apr 18 2000

a(n) = F(n, 2), the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Define c(2n) = -A001108(n), c(2n+1) = -A001108(n+1) and d(2n) = d(2n+1) = A001652(n); then ((-1)^n)*(c(n) + d(n)) = a(n). [Proof given by Max Alekseyev.] - Creighton Dement, Jul 21 2005

a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - Lekraj Beedassy, Sep 03 2006

a(n) = (b(n+1) + b(n-1))/n where {b(n)} is the sequence A006645. - Sergio Falcon, Nov 22 2006

From Miklos Kristof, Mar 19 2007: (Start)

Let F(n) = a(n) = Pell numbers, L(n) = A002203 = companion Pell numbers (A002203):

For a >= b and odd b,  F(a+b) + F(a-b) = L(a)*F(b).

For a >= b and even b, F(a+b) + F(a-b) = F(a)*L(b).

For a >= b and odd b,  F(a+b) - F(a-b) = F(a)*L(b).

For a >= b and even b, F(a+b) - F(a-b) = L(a)*F(b).

F(n+m) + (-1)^m*F(n-m) = F(n)*L(m).

F(n+m) - (-1)^m*F(n-m) = L(n)*F(m).

F(n+m+k) + (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = F(n)*L(m)*L(k).

F(n+m+k) - (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = L(n)*L(m)*F(k).

F(n+m+k) + (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = L(n)*F(m)*L(k).

F(n+m+k) - (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = 8*F(n)*F(m)*F(k). (End)

a(n+1)*a(n) = 2*Sum_{k=0..n} a(k)^2 (a similar relation holds for A001333). - Creighton Dement, Aug 28 2007

a(n+1) = Sum_{k=0..n} binomial(n+1,2k+1) * 2^k = Sum_{k=0..n} A034867(n,k) * 2^k = (1/n!) * Sum_{k=0..n} A131980(n,k) * 2^k. - Tom Copeland, Nov 30 2007

Equals row sums of unsigned triangle A133156. - Gary W. Adamson, Apr 21 2008

a(n) (n >= 3) is the determinant of the (n-1) X (n-1) tridiagonal matrix with diagonal entries 2, superdiagonal entries 1 and subdiagonal entries -1. - Emeric Deutsch, Aug 29 2008

a(n) = 5*a(n-2) + 2*a(n-3), a(n) = 6*a(n-2) - a(n-4). - Mohamed Bouhamida, Sep 04 2008

a(n) = A000045(n) + Sum_{k=1..n-1} A000045(k)*a(n-k). - Roger L. Bagula and Gary W. Adamson, Sep 07 2008

From Hieronymus Fischer, Jan 02 2009: (Start)

fract((1+sqrt(2))^n)) = (1/2)*(1 + (-1)^n) - (-1)^n*(1+sqrt(2))^(-n) = (1/2)*(1 + (-1)^n) - (1-sqrt(2))^n.

See A001622 for a general formula concerning the fractional parts of powers of numbers x > 1, which satisfy x - x^(-1) = floor(x).

a(n) = round((1+sqrt(2))^n) for n > 0. (End)

a(n)=((4+sqrt(18))*(1+sqrt(2))^n) + (4-sqrt(18))*(1-sqrt(2))^n)/4 offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

If p[i] = Fibonacci(i) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1] when i<=j, A[i,j]=-1 when i=j+1, and A[i,j]=0 otherwise, then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010

a(n) = 3*a(n-1) - a(n-2) - a(n-3), n > 2. - Gary Detlefs, Sep 09 2010

a(n) = 2*(a(2k-1) + a(2k))*a(n-2k) - a(n-4k).

  a(n) = 2*(a(2k) + a(2k+1))*a(n-2k-1) + a(n-4k-2). - Charlie Marion, Apr 13 2011

G.f.: x/(1 - 2*x - x^2) = sqrt(2)*G(0)/4; G(k) = ((-1)^k) - 1/(((sqrt(2) + 1)^(2*k)) - x*((sqrt(2) + 1)^(2*k))/(x + ((sqrt(2) - 1)^(2*k + 1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011

In general, for n > k, a(n) = a(k+1)*a(n-k) + a(k)*a(n-k-1). See definition of Pell numbers and the formula for Sep 04 2008. - Charlie Marion, Jan 17 2012

Sum{n>=1} (-1)^(n-1)/(a(n)*a(n+1)) = sqrt(2) - 1. - Vladimir Shevelev, Feb 22 2013

From Vladimir Shevelev, Feb 24 2013: (Start)

(1) Expression a(n+1) via a(n): a(n+1) = a(n) + sqrt(2*a^2(n) + (-1)^n);

(2) a(n+1)^2 - a(n)*a(n+2) = (-1)^n;

(3) Sum_{k=1..n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);

(4) a(n)/a(n+1) = sqrt(2) - 1 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)

a(-n) = -(-1)^n * a(n). - Michael Somos, Jun 01 2013

G.f.: G(0)/(2+2*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

a(n) = Sum_{r=0..n-1} Sum_{k=0..n-r-1} binomial(r+k,k)*binomial(k,n-k-r-1). - Peter Luschny, Nov 16 2013

a(n) = Sum_{k=1,3,5,...<=n} C(n,k)*2^((k-1)/2). - Vladimir Shevelev, Feb 06 2014

a(2n) = 2*a(n)*(a(n-1) + a(n)). - John Blythe Dobson, Mar 08 2014

a(k*n) = a(k)*a(k*n-k+1) + a(k-1)*a(k*n-k). - Charlie Marion, Mar 27 2014

a(k*n) = 2*a(k)*(a(k*n-k)+a(k*n-k-1)) + (-1)^k*a(k*n-2k). - Charlie Marion, Mar 30 2014

a(n+1) = (1+sqrt(2))*a(n) + (1-sqrt(2))^n. - Art DuPre, Apr 04 2014

a(n+1) = (1-sqrt(2))*a(n) + (1+sqrt(2))^n. - Art DuPre, Apr 04 2014

a(n) = F(n) + Sum_{k=1..n} F(k)*a(n-k), n >= 0 where F(n) the Fibonacci numbers A000045. - Ralf Stephan, May 23 2014

a(n) = round(sqrt(a(2n) + a(2n-1)))/2. - Richard R. Forberg, Jun 22 2014

a(n) = Product_{k divides n} A008555(k). - Tom Edgar, Jan 28 2015

a(n+k)^2 - A002203(k)*a(n)*a(n+k) + (-1)^k*a(n)^2 = (-1)^n*a(k)^2. - Alexander Samokrutov, Aug 06 2015

a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1)) for n >= 2. - Peter Luschny, Dec 17 2015

a(n+1) = Sum_{k=0..n} binomial(n,k)*2^floor(k/2). - Tony Foster III, May 07 2017

a(n) = exp((i*Pi*n)/2)*sinh(n*arccosh(-i))/sqrt(2). - Peter Luschny, Mar 07 2018

From Rogério Serôdio, Mar 30 2018: (Start)

Some properties:

(1) a(n)^2 - a(n-2)^2 = 2*a(n-1)*(a(n) + a(n-2)) (see A005319);

(2) a(n-k)*a(n+k) = a(n)^2 + (-1)^(n+k+1)*a(k)^2;

(3) Sum_{k=2..n+1} a(k)*a(k-1) = a(n+1)^2 if n is odd, else a(n+1)^2 - 1 if n is even;

(4) a(n) - a(n-2*k+1) = (A077444(k) - 1)*a(n-2*k+1) + a(n-4*k+2);

(5) Sum_{k=n..n+9} a(k) = 41*A001333(n+5). (End)

From Kai Wang, Dec 30 2019: (Start)

a(m+r)*a(n+s) - a(m+s)*a(n+r) = -(-1)^(n+s)*a(m-n)*a(r-s).

a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A002203(m+n+r+s) - (-1)^(n+s)*A002203(m-n)*A002203(r-s))/8.

A002203(m+r)*A002203(n+s) - A002203(m+s)*A002203(n+r) = (-1)^(n+s)*8*a(m-n)*a(r-s).

A002203(m+r)*A002203(n+s) - 8*a(m+s)*a(n+r) = (-1)^(n+s)*A002203(m-n)*A002203(r-s).

A002203(m+r)*A002203(n+s) + 8*a(m+s)*a(n+r) = 2*A002203(m+n+r+s)+ (-1)^(n+s)*8*a(m-n)*a(r-s). (End)

From Kai Wang, Jan 12 2020: (Start)

a(n)^2 - a(n+1)*a(n-1) = (-1)^(n-1).

a(n)^2 - a(n+r)*a(n-r) = (-1)^(n-r)*a(r)^2.

a(m)*a(n+1) - a(m+1)*a(n) = (-1)^n*a(m-n).

a(m-n) = (-1)^n (a(m)*A002203(n) - A002203(m)*a(n))/2.

a(m+n) = (a(m)*A002203(n) + A002203(m)*a(n))/2.

A002203(n)^2 - A002203(n+r)*A002203(n-r) = (-1)^(n-r-1)*8*a(r)^2.

A002203(m)*A002203(n+1) - A002203(m+1)*A002203(n) = (-1)^(n-1)*8*a(m-n).

A002203(m-n) = (-1)^(n)*(A002203(m)*A002203(n) - 8*a(m)*a(n) )/2.

A002203(m+n) = (A002203(m)*A002203(n) + 8*a(m)*a(n) )/2. (End)

From Kai Wang, Mar 03 2020: (Start)

Sum_{m>=1} arctan(2/a(2*m+1)) = arctan(1/2).

Sum_{m>=2} arctan(2/a(2*m+1)) = arctan(1/12).

In general, for n > 0,

Sum_{m>=n} arctan(2/a(2*m+1)) = arctan(1/a(2*n)). (End)

a(n) = (A001333(n+3*k) + (-1)^(k-1)*A001333(n-3*k)) / (20*A041085(k-1)) for any k>=1. - Paul Curtz, Jun 23 2021

Sum_{i=0..n} a(i)*J(n-i) = (a(n+1) + a(n) - J(n+2))/2 for J(n) = A001045(n). - Greg Dresden, Jan 05 2022

G.f. = x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 70*x^6 + 169*x^7 + 408*x^8 + 985*x^9 + ...

A000129 := proc(n) option remember; if n <=1 then n; else 2*procname(n-1)+procname(n-2); fi; end;

a:= n-> (<<2|1>, <1|0>>^n)[1, 2]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 01 2008

A000129 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1)):

seq(simplify(A000129(n)), n=0..31); # Peter Luschny, Dec 17 2015

CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] (* Stefan Steinerberger, Apr 08 2006 *)

Expand[Table[((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)

LinearRecurrence[{2, 1}, {0, 1}, 60] (* Harvey P. Dale, Jan 04 2012 *)

a[ n_] := With[ {s = Sqrt@2}, ((1 + s)^n - (1 - s)^n) / (2 s)] // Simplify; (* Michael Somos, Jun 01 2013 *)

Table[Fibonacci[n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, May 08 2016 *)

Fibonacci[Range[0, 20], 2] (* Eric W. Weisstein, Sep 30 2017 *)

a[ n_] := ChebyshevU[n - 1, I] / I^(n - 1); (* Michael Somos, Oct 30 2021 *)

(PARI) default(realprecision, 2000); for (n=0, 4000, a=contfracpnqn(vector(n, i, 1+(i>1)))[2, 1]; if (a > 10^(10^3 - 6), break); write("b000129.txt", n, " ", a)); \\ Harry J. Smith, Jun 12 2009

(PARI) {a(n) = imag( (1 + quadgen( 8))^n )}; /* Michael Somos, Jun 01 2013 */

(PARI) {a(n) = if( n<0, -(-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [2, 1]}; /* Michael Somos, Jun 01 2013 */

(PARI) a(n)=([2, 1; 1, 0]^n)[2, 1] \\ Charles R Greathouse IV, Mar 04 2014

(PARI) {a(n) = polchebyshev(n-1, 2, I) / I^(n-1)}; /* Michael Somos, Oct 30 2021 */

(Sage) [lucas_number1(n, 2, -1) for n in range(30)]  # Zerinvary Lajos, Apr 22 2009

(Haskell)

a000129 n = a000129_list !! n

a000129_list = 0 : 1 : zipWith (+) a000129_list (map (2 *) $ tail a000129_list)

-- Reinhard Zumkeller, Jan 05 2012, Feb 05 2011

(Maxima)

a[0]:0$

a[1]:1$

a[n]:=2*a[n-1]+a[n-2]$

A000129(n):=a[n]$

makelist(A000129(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

(Maxima) makelist((%i)^(n-1)*ultraspherical(n-1, 1, -%i), n, 0, 24), expand; /* Emanuele Munarini, Mar 07 2018 */

(Magma) [0] cat [n le 2 select n else 2*Self(n-1) + Self(n-2): n in [1..35]]; // Vincenzo Librandi, Aug 08 2015

(GAP) a := [0, 1];; for n in [3..10^3] do a[n] := 2 * a[n-1] + a[n-2]; od; A000129 := a; # Muniru A Asiru, Oct 16 2017

(Python)

from itertools import islice

def A000129_gen(): # generator of terms

    a, b = 0, 1

    yield from [a, b]

    while True:

        a, b = b, a+2*b

        yield b

A000129_list = list(islice(A000129_gen(), 20)) # Chai Wah Wu, Jan 11 2022

Partial sums of A001333.

2nd row of A172236.

a(n) = A054456(n-1, 0), n>=1 (first column of triangle).

Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409.

Cf. A175181 (Pisano periods), A214028 (Entry points), A214027 (number of zeros in a fundamental period).

A077985 is a signed version.

INVERT transform of Fibonacci numbers (A000045).

Cf. A038207.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Cf. A034867, A131980, A133156, A143808, A135387, A153346, A001622, A006497, A014176 (growth power), A098316, A154325, A021083, A243399, A008555.

Cf. A048739.

Cf. A073133.

Cf. A041085.

Sequence in context: A048624 A176981 A215936 * A077985 A215928 A054198

Adjacent sequences:  A000126 A000127 A000128 * A000130 A000131 A000132

nonn,easy,core,cofr,nice,frac

N. J. A. Sloane

approved