0,4

Also (0,1)-strings such that all maximal blocks of 1's have even length and all maximal blocks of 0's have odd length.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problems 2.4.3, 2.4.13).

Table of n, a(n) for n=0..44.

Index entries for linear recurrences with constant coefficients, signature (0, 2, 1, -1).

a(n) = Sum_{j=0..n+1} binomial(n-j+1, 3*j-n+1).

a(n) = 2*a(n-2) + a(n-3) - a(n-4).

G.f.: -(x^2-x-1)/(x^4-x^3-2*x^2+1). More generally, g.f. for the number of compositions of n such that two adjacent parts are not equal modulo p is 1/(1-Sum_{i=1..p} x^i/(1+x^i-x^p)).

G.f.: W(0)/(2*x^2) -1/x^2, where W(k) = 1 + 1/(1 - x*(k - x)/( x*(k+1 - x) - 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

From Joerg Arndt, Oct 27 2012: (Start)

The 11 such compositions of n=7 are

[ 1] 1 2 1 2 1

[ 2] 1 6

[ 3] 2 1 4

[ 4] 2 3 2

[ 5] 2 5

[ 6] 3 4

[ 7] 4 1 2

[ 8] 4 3

[ 9] 5 2

[10] 6 1

[11] 7

The 16 such compositions of n=8 are

[ 1] 1 2 1 4

[ 2] 1 2 3 2

[ 3] 1 2 5

[ 4] 1 4 1 2

[ 5] 1 4 3

[ 6] 1 6 1

[ 7] 2 1 2 1 2

[ 8] 2 1 2 3

[ 9] 2 1 4 1

[10] 2 3 2 1

[11] 3 2 1 2

[12] 3 2 3

[13] 3 4 1

[14] 4 1 2 1

[15] 5 2 1

[16] 8

(End)

LinearRecurrence[{0, 2, 1, -1}, {1, 1, 1, 3}, 50] (* Harvey P. Dale, Feb 26 2012 *)

Join[{1}, Table[Sum[ Binomial[n-j+1, 3j-n+1], {j, 0, n-1}], {n, 50}]] (* Harvey P. Dale, Feb 26 2012 *)

(PARI) x='x+O('x^66); Vec(-(x^2-x-1)/(x^4-x^3-2*x^2+1)) \\ Joerg Arndt, May 13 2013

Cf. A003242, A062201-A062203.

Sequence in context: A154028 A157793 A096375 * A114208 A014686 A053090

Adjacent sequences: A062197 A062198 A062199 * A062201 A062202 A062203

nonn,easy

Vladeta Jovovic, Jun 13 2001

approved