A094865 Expansion of x^3/((1-3*x+x^2)*(1-5*x+5*x^2)).

0, 0, 0, 1, 8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, 8844448, 32256553, 117378336, 426440955, 1547491404, 5610955132, 20332248992, 73645557469, 266668876540, 965384509651, 3494279574288, 12646311635088, 45764967830976, 165605867248465

Offset: 0

Author

Herbert Kociemba, Jun 15 2004

Keywords

• nonn
• easy

Comments

With a different offset, number of sequences (s(0), s(1), ..., s(2k+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2k+1, with s(0) = 1 and s(2n+1) = 8.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Roger L. Bagula and Gary W. Adamson, Comments on this sequence
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Crossrefs

Cf. A005024 is a truncated version.

Programs

Mathematica

CoefficientList[Series[x^3/((1-3x+x^2)(1-5x+5x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-21,20,-5},{0,0,0,1},30] (* Harvey P. Dale, Jun 07 2014 *)

PARI

x='x+O('x^66); concat([0,0,0],Vec(x^3/((1-3*x+x^2)*(1-5*x+5*x^2)))) \\ Joerg Arndt, May 01 2013

Formula

a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(4*r*Pi/5)*(2*cos(r*Pi/10))^(2*n+1).

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).

a(n) = A093129(n)/2 - A122367(n)/2. - R. J. Mathar, Jun 24 2011

a(n) = 2^(-2-n)*(-(3-sqrt(5))^n*(-1+sqrt(5)) + (5-sqrt(5))^n*(1+sqrt(5)) - (1+sqrt(5))*(3+sqrt(5))^n + (-1+sqrt(5))*(5+sqrt(5))^n)/sqrt(5). - Colin Barker, Apr 27 2016

Extensions

Edited by N. J. A. Sloane, Aug 09 2008