A132723 Binomial transform of A132429.

3, 4, 4, 0, -8, -16, -16, 0, 32, 64, 64, 0, -128, -256, -256, 0, 512, 1024, 1024, 0, -2048, -4096, -4096, 0, 8192, 16384, 16384, 0, -32768, -65536, -65536, 0, 131072, 262144, 262144, 0, -524288, -1048576, -1048576

Offset: 0

Author

Paul Curtz, Nov 16 2007

Keywords

• sign

Comments

Sequence is identical to its fourth differences.

Links

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

Programs

Magma

[3] cat [n le 2 select 4 else 2*(Self(n-1) - Self(n-2)): n in [1..40]]; // G. C. Greubel, Feb 14 2021

Mathematica

Join[{3},LinearRecurrence[{2,-2},{4,4},50]] (* Harvey P. Dale, Mar 06 2014 *)
Table[If[n<2, n+3, 2*((1+I)^(n-1) + (1-I)^(n-1))], {n,0,40}] (* G. C. Greubel, Feb 14 2021 *)

Sage

def A132723(n): return n+3 if (n<2) else 2*( (1+i)^(n-1) + (1-i)^(n-1) )
[A132723(n) for n in (0..40)] # G. C. Greubel, Feb 14 2021

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n positive. For a(0)=3,a(1)=a(2)=4,a(3)=0.

From R. J. Mathar, Apr 02 2008: (Start)

O.g.f.: 1 + 2/(1 -2*x +2*x^2).

a(n) = 2*a(n-1) - 2*a(n-2) if n>2. (End)

E.g.f.: 1 + 2*sqrt(2)*exp(x)*sin(x + Pi/4). - G. C. Greubel, Feb 14 2021

Extensions

More terms from R. J. Mathar, Apr 02 2008