A131259 a(2*n) = A000217(n), a(2*n+1) = -2*A000217(n).

0, 0, 1, -2, 3, -6, 6, -12, 10, -20, 15, -30, 21, -42, 28, -56, 36, -72, 45, -90, 55, -110, 66, -132, 78, -156, 91, -182, 105, -210, 120, -240, 136, -272, 153, -306, 171, -342, 190, -380, 210, -420, 231, -462, 253, -506, 276, -552, 300, -600, 325, -650, 351, -702, 378, -756, 406, -812, 435, -870, 465, -930, 496, -992

Offset: 0

Author

Paul Curtz, Sep 28 2007

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000217, A129819.

Programs

Magma

A131259:= func< n | (2*(n mod 2) +2*((n+1) mod 2)*n -((n mod 2) -(-1)^n)*n^2 )/8 >;
[A131259(n): n in [0..70]]; // G. C. Greubel, Jan 22 2025

Maple

A000217 := proc(n) n*(n+1)/2 ; end: A131259 := proc(n) if n mod 2 = 0 then A000217(n/2) ; else -2*A000217((n-1)/2) ; fi ; end: seq(A131259(n),n=0..60) ; # R. J. Mathar, Oct 26 2007

Mathematica

CoefficientList[Series[x^2*(1 - 2*x)/(1 - x^2)^3, {x,0,35}], x] (* Georg Fischer, Nov 16 2022 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,0,1,-2,3,-6},70] (* Harvey P. Dale, Jul 07 2024 *)

Python

def A131259(n): return (2*(n%2) +2*((n+1)%2)*n -(3*(n%2)-1)*n**2)//8
print([A131259(n) for n in range(71)]) # G. C. Greubel, Jan 22 2025

Formula

O.g.f.: ogf217(x^2) - 2*x*ogf217(x^2) = x^2/(1-x^2)^3 - 2*x^3/(1-x^2) = x^2*(1 - 2*x) / (1 - x^2)^3. - Georg Fischer, Nov 16 2022

From G. C. Greubel, Jan 22 2025: (Start)

a(n) = (1/16)*(2*(1-(-1)^n) + 2*(1+(-1)^n)*n - (1-3*(-1)^n)*n^2).

E.g.f.: (1/8)*(x-2)*(x*cosh(x) - (1+2*x)*sinh(x)). (End)

Extensions

More terms from R. J. Mathar, Oct 26 2007