A117902 Expansion of (1-x^2-2x^3)/(1-4x^3).

1, 0, -1, 2, 0, -4, 8, 0, -16, 32, 0, -64, 128, 0, -256, 512, 0, -1024, 2048, 0, -4096, 8192, 0, -16384, 32768, 0, -65536, 131072, 0, -262144, 524288, 0, -1048576, 2097152, 0, -4194304, 8388608, 0, -16777216, 33554432, 0, -67108864, 134217728, 0, -268435456, 536870912, 0, -1073741824

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• sign

Comments

Row sums of number triangle A117901.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Crossrefs

Cf. A117901, A133851.

Programs

Magma

[1] cat [n le 3 select (-1)^(n-1)*(n-1) else 4*Self(n-3): n in [1..50]]; // G. C. Greubel, Oct 09 2021

Mathematica

LinearRecurrence[{0,0,4}, {1,0,-1,2}, 50] (* G. C. Greubel, Oct 09 2021 *)

Sage

def A133851(n): return 4^(n/3) if (n%3==0) else 0
def A117902(n): return bool(n==0)/2 + 2*A133851(n-3) - A133851(n-2)
[A117902(n) for n in (0..50)] # G. C. Greubel, Oct 09 2021

Formula

a(n) = 0^n/2 - (2^(2*n/3)/12)*( 2*cos((2*n+1)*Pi*n/3) + 2*sqrt(3)*sin((2*n+1)*Pi*n/3) -(2^(2/3) + 8)*cos(2*Pi*n/3) - 2^(1/6)*sqrt(6)*sin(2*Pi*n/3) + 2^(2/3) - 2 ).

a(n) = (1/2)*[n=0] + 2*A133851(n-3) - A133851(n-2). - G. C. Greubel, Oct 09 2021