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A117761 Expansion of x*(1 + x^2 + x^4)/(1 - x - x^3 - x^5 - x^7). 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified April 17 18:43 EDT 2024. Contains 371765 sequences. (Running on oeis4.)