|
| |
|
|
A057086
|
|
Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.
|
|
10
|
|
|
|
1, 10, 90, 800, 7100, 63000, 559000, 4960000, 44010000, 390500000, 3464900000, 30744000000, 272791000000, 2420470000000, 21476790000000, 190563200000000, 1690864100000000, 15003009000000000, 133121449000000000, 1181184400000000000, 10480629510000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.
The characteristic roots are rp(m) := (m+sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)) is the Binet form of these m-sequences.
|
|
|
LINKS
|
Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=10, q=-10.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-10).
|
|
|
FORMULA
|
a(n) = 10*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A057080(k)*10^k, a(2*k+1) = A001090(k)*10^(k+1).
G.f.: 1/(1-10*x+10*x^2).
a(n) = Sum_[k, 0<=k<=n} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008
a(n) = -(1/30)*[5-sqrt(15)]^(n+1)*sqrt(15)+(1/30)*sqrt(15)*[5+sqrt(15)]^(n+1), with n>=0. - Paolo P. Lava, Nov 20 2008
|
|
|
MATHEMATICA
|
Join[{a=1, b=10}, Table[c=10*b-10*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)
|
|
|
PROG
|
(Sage) [lucas_number1(n, 10, 10) for n in xrange(1, 20)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
|
|
|
CROSSREFS
|
Cf. A030240, A057084.
Sequence in context: A199940 A004985 A164552 * A092420 A010579 A010576
Adjacent sequences: A057083 A057084 A057085 * A057087 A057088 A057089
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Wolfdieter Lang Aug 11 2000
|
|
|
STATUS
|
approved
|
| |
|
|
