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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 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:=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

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