A094865 - OEIS

%I #33 Feb 12 2022 17:51:33

%S 0,0,0,1,8,43,196,820,3264,12597,47652,177859,657800,2417416,8844448,

%T 32256553,117378336,426440955,1547491404,5610955132,20332248992,

%U 73645557469,266668876540,965384509651,3494279574288,12646311635088,45764967830976,165605867248465

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

%C 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.

%H Colin Barker, <a href="/A094865/b094865.txt">Table of n, a(n) for n = 0..1000</a>

%H Roger L. Bagula and Gary W. Adamson, <a href="/A094865/a094865.txt">Comments on this sequence</a>

%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).

%F 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).

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

%F a(n) = A093129(n)/2 - A122367(n)/2. - _R. J. Mathar_, Jun 24 2011

%F 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

%t 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 *)

%o (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

%Y Cf. A005024 is a truncated version.

%K nonn,easy

%O 0,5

%A _Herbert Kociemba_, Jun 15 2004

%E Edited by _N. J. A. Sloane_, Aug 09 2008