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A217988
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Binomial transform of A215495(n).
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2
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1, 2, 4, 10, 26, 66, 160, 372, 840, 1864, 4096, 8944, 19424, 41952, 90112, 192576, 409728, 868480, 1835008, 3866368, 8125952, 17038848, 35651584, 74449920, 155191296, 322963456, 671088640, 1392504832, 2885672960, 5972680704, 12348030976, 25501384704
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OFFSET
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0,2
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COMMENTS
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Like any other sequence with a linear recurrence with constant coefficients, this sequence is periodic if read modulo some constant m. These Pisano period lengths for m>=1 are 1, 1, 8, 1, 20, 8, 168, 1, 24, 20, 440, 8, 156, 168, 40, 1, 272, 24, 1368, 20, ... [Curtz's comment reformulated and extended by R. J. Mathar, Oct 23 2012]
Let b(n) = a(n+1)-2*a(n), then b(n+3)-2*b(n+2) = A009545(n+2). - edited by Michel Marcus, Apr 24 2018
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LINKS
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FORMULA
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a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4) with n > 5.
G.f.: (1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 22 2012
a(n) = 2^(n-3)*(3*n+2)+((1+i)^n+(1-i)^n)/4, where i=sqrt(-1) and n>1, with a(0)=1, a(1)=2.
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EXAMPLE
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a(n) and successive differences:
1, 2, 4, 10, 26, 66, 160, 372, 840, 1864, 4096, ...
1, 2, 6, 16, 40, 94, 212, 468, 1024, ...
1, 4, 10, 24, 54, 118, 256, ...
3, 6, 14, 30, 64, ...
3, 8, 16, ...
5, 8, ...
3, ...
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MATHEMATICA
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a[n_] := Sum[ Binomial[n, k]*If[ OddQ[k], k, k/2 + Boole[ Mod[k, 4] == 0]], {k, 0, n}]; Table[ a[n], {n, 0, 31}] (* Jean-François Alcover, Oct 17 2012 *)
CoefficientList[Series[(1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2 * (1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{6, -14, 16, -8}, {1, 2, 4, 10, 26, 66}, 40] (* Harvey P. Dale, Aug 14 2018 *)
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PROG
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(Magma) I:=[1, 2, 4, 10, 26, 66]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012
(PARI) x='x+O('x^30); Vec((1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2))) \\ G. C. Greubel, Apr 23 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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