A131885 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) for n >= 4 starting with a(0) = 1, a(1) = 2, a(2) = 4, and a(3) = 6.

1, 2, 4, 6, 8, 12, 24, 56, 128, 272, 544, 1056, 2048, 4032, 8064, 16256, 32768, 65792, 131584, 262656, 524288, 1047552, 2095104, 4192256, 8388608, 16781312, 33562624, 67117056, 134217728, 268419072, 536838144, 1073709056, 2147483648, 4295032832, 8590065664

Offset: 0

Author

Paul Curtz, Oct 25 2007

Keywords

• nonn

Links

• Karl Dilcher and Maciej Ulas, Divisibility and Arithmetic Properties of a Class of Sparse Polynomials, arXiv:2008.13475 [math.NT], 2020. See Table 1, 2nd column, p. 3.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Programs

Mathematica

Join[{1},LinearRecurrence[{4,-6,4},{2,4,6},60]] (* Harvey P. Dale, Jul 07 2011 *)

Formula

Binomial transform of 1, 1, 1, -1.

G.f.: (-1 + 2*x - 2*x^2 + 2*x^3)/(2*x - 1)/(2*x^2 - 2*x + 1). - R. J. Mathar, Nov 14 2007

a(n) = 2*A038504(n) for n > 0. - R. J. Mathar, Jul 17 2009

G.f.: 1/2*(1 - 1/(2*x-1) + x*Q(0)/(1-x)), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/Q(k+1) )) (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013

a(n) = Sum_{j=0..n} binomial(n, j)*(-1)^binomial(j, 3); this is the case m=3 and z=-1 of f(m,n)(z) = Sum_{j=0..n} binomial(n, j)*z^binomial(j, m). See Dilcher and Ulas. - Michel Marcus, Sep 01 2020

Extensions

More terms from Harvey P. Dale, Jul 07 2011