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A094865 Expansion of x^3/((1-3*x+x^2)*(1-5*x+5*x^2)). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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
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.
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
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 *)
PROG
(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
CROSSREFS
Cf. A005024 is a truncated version.
Sequence in context: A055853 A137748 A005024 * A122880 A171479 A227209
KEYWORD
nonn,easy
AUTHOR
Herbert Kociemba, Jun 15 2004
EXTENSIONS
Edited by N. J. A. Sloane, Aug 09 2008
STATUS
approved

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Last modified March 29 03:39 EDT 2024. Contains 371264 sequences. (Running on oeis4.)