A117761 Expansion of x*(1 + x^2 + x^4)/(1 - x - x^3 - x^5 - x^7).

0, 1, 1, 2, 3, 5, 8, 12, 20, 32, 51, 82, 131, 210, 336, 538, 862, 1380, 2210, 3539, 5667, 9075, 14532, 23271, 37265, 59674, 95559, 153023, 245043, 392399, 628367, 1006234, 1611330, 2580299, 4131955, 6616695, 10595627, 16967279, 27170507, 43509419

Offset

0,4

Comments

Previous name: First entry of the vector (M^n)w, where M is the 7x7 matrix [[0,1,0,0,0,0,0], [0,0,1,0,0,0,0], [0,0,0,1,0,0,0], [0,0,0,0,1,0,0], [0,0,0,0,0,1,0], [0,0,0,0,0,0,1], [1,0,1,0,1,0,1]] and w is the column vector [0,1,1,2,3,5,8].

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7).

G.f.: (x + x^3 + x^5)/(1 - x - x^3 - x^5 - x^7). - Joel B. Lewis, Nov 14 2012

Maple

with(linalg): M:=matrix(7, 7, [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1]):
w[0]:=matrix(7, 1, [0, 1, 1, 2, 3, 5, 8]): for n from 1 to 40 do
w[n]:=multiply(M, w[n-1]) od: seq(w[n][1, 1], n=0..40);

Mathematica

M = {{0, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1}};
w[1] = {0, 1, 1, 2, 3, 5, 8}; w[n_]:= w[n]= M.w[n-1];
A117761[n_]:= w[n][[1]];
Table[A117761[n], {n, 50}]
LinearRecurrence[{1, 0, 1, 0, 1, 0, 1}, {0, 1, 1, 2, 3, 5, 8}, 50] (* Harvey P. Dale, Oct 06 2017 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x*(1+x^2+x^4)/(1-x-x^3-x^5-x^7) )); // G. C. Greubel, Jul 21 2023
(SageMath)
@CachedFunction
def a(n): # a = A117761
    if n<8: return fibonacci(n) - int(n==7)
    else: return a(n-1) + a(n-3) + a(n-5) + a(n-7)
[a(n) for n in range(51)] # G. C. Greubel, Jul 21 2023

Crossrefs

Cf. A117760.

Sequence in context: A170805 A013984 A107479 * A274376 A018149 A152909

Adjacent sequences: A117758 A117759 A117760 * A117762 A117763 A117764

Keyword

nonn,easy

Author

Roger L. Bagula, Apr 14 2006

Extensions

Edited by N. J. A. Sloane, Apr 30 2006

New name from Joel B. Lewis, Nov 14 2012

Status

approved