A217988 Binomial transform of A215495(n).
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
Offset: 0
Examples
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, ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).
Programs
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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
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Mathematica
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 *) -
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
Formula
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.
Comments