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A218009 Binomial transform of A212831(n). 2
0, 1, 4, 12, 30, 70, 160, 364, 824, 1848, 4096, 8976, 19488, 42016, 90112, 192448, 409472, 868224, 1835008, 3866880, 8126976, 17039872, 35651584, 74447872, 155187200, 322959360, 671088640, 1392513024, 2885689344, 5972697088, 12348030976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Companion to A217988.

Considering a(n+1) - 2*a(n) = 1,2,4,6,10,20,44,96,200,... = b(n), is

b(n+3) - 2*b(n+2) = -2,-2,0,4,8,8,0,-16,-32,-32,0,... = -A009545(n+2).

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).

FORMULA

G.f.: x*(1-2*x+2*x^2-2*x^3+2*x^4)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 18 2012

a(n) = 2^(n-3)*(3*n+2)+(1+i)^(n-4)+(1-i)^(n-4), where i=sqrt(-1) and n>1, with a(0)=0, a(1)=1. - Bruno Berselli, Oct 18 2012

a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -8*a(n-4).

MATHEMATICA

a[n_] := Sum[(1/4)*Binomial[n, k]*((-(1 + (-1)^k))*(-1 + (-1)^Floor[k/2]) - (-3 + (-1)^k)*k), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 18 2012 *)

CoefficientList[Series[x*(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4)/((1 - 2*x)^2*(1 - 2*x + 2*x^2)), {x, 0, 40}], x]; // Vincenzo Librandi, Dec 15 2012

PROG

(MAGMA) I:=[0, 1, 4, 12, 30, 70]; [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

CROSSREFS

Cf. A214282, A214283.

Sequence in context: A036388 A100691 A000298 * A249914 A006802 A068055

Adjacent sequences:  A218006 A218007 A218008 * A218010 A218011 A218012

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Oct 18 2012

STATUS

approved

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Last modified March 5 11:06 EST 2021. Contains 341823 sequences. (Running on oeis4.)