A131666 First differences of (A113405 prefixed with a 0).

0, 0, 1, 1, 2, 3, 7, 14, 29, 57, 114, 227, 455, 910, 1821, 3641, 7282, 14563, 29127, 58254, 116509, 233017, 466034, 932067, 1864135, 3728270, 7456541, 14913081, 29826162, 59652323, 119304647, 238609294, 477218589, 954437177, 1908874354

Offset: 0

Author

Paul Curtz, Sep 14 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Programs

GAP

a:=[0,0,1,1];; for n in [5..35] do a[n]:=2*a[n-1]-a[n-3]+2*a[n-4]; od; a; # G. C. Greubel, Nov 21 2019

Magma

R:=PowerSeriesRing(Integers(), 35); [0,0] cat Coefficients(R!( x^2*(1-x)/((1+x)*(1-2*x)*(1-x+x^2)) )); // G. C. Greubel, Nov 21 2019

Maple

seq(coeff(series(x^2*(1-x)/((1+x)*(1-2*x)*(1-x+x^2)), x, n+1), x, n), n = 0..35); # G. C. Greubel, Nov 21 2019

Mathematica

Table[(2*(-1)^n +2^n -3*((-1)^Floor[(n+1)/3] +(-1)^Floor[(n+2)/3])/2)/9, {n, 0, 35}] (* G. C. Greubel, Nov 21 2019 *)

PARI

my(x='x+O('x^35)); concat([0,0], Vec(x^2*(1-x)/((1+x)*(1-2*x)*(1-x+x^2)))) \\ G. C. Greubel, Nov 21 2019

Sage

def A131666_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1-x)/((1+x)*(1-2*x)*(1-x+x^2))).list()
A131666_list(35) # G. C. Greubel, Nov 21 2019

Formula

a(n+1) - 2*a(n) = 0, 1, -1, 0, -1, 1, ... is the hexaperiodic A092220 (corrected Sep 24 2007).

O.g.f.: x^2*(1-x)/((1+x)*(1-2*x)*(1-x+x^2)). - R. J. Mathar, Nov 23 2007

a(n) = Sum_{k=0..n-2} A001045(n-k+1)*binomial(k,n-k-2). - Paul Barry, Apr 22 2009

a(n) = (1/18)*( 2^(n+1) + 4*(-1)^n - 3*((-1)^floor((n+1)/3) + (-1)^floor((n+2)/3)) ). - G. C. Greubel, Nov 21 2019