A218009 Binomial transform of A212831(n).

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, 25501351936, 52613316608

Offset: 0

Author

Paul Curtz, Oct 18 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A212831, A214282, A214283, A217988, A009545.

Programs

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

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 *)

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).