A027656 Expansion of 1/(1-x^2)^2 (included only for completeness - the policy is always to omit the zeros from such sequences).

1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 0, 19, 0, 20, 0, 21, 0, 22, 0, 23, 0, 24, 0, 25, 0, 26, 0, 27, 0, 28, 0, 29, 0, 30, 0, 31, 0, 32, 0, 33, 0, 34, 0, 35, 0, 36, 0, 37, 0, 38, 0, 39, 0, 40, 0, 41, 0, 42, 0, 43, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of nonnegative integer solutions to the equation x+y+z=n such that x+y=z. - Geoffrey Critzer, Jul 12 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Crossrefs

Cf. A008619, A008805, A045891, A059841, A142150.

Programs

Magma

[(n+2)*(1+(-1)^n)/4: n in [0..75]]; // Vincenzo Librandi, Apr 02 2011

Mathematica

CoefficientList[Series[1/(1-x^2)^2,{x,0,100}],x] (* Geoffrey Critzer, Jul 12 2013 *)

PARI

a(n)=if(n%2,0,n/2+1) \\ Charles R Greathouse IV, Jan 18 2012

SageMath

[(n+2)*((n+1)%2)/2 for n in (0..80)] # G. C. Greubel, Aug 01 2022

Formula

From Paul Barry, May 27 2003: (Start)

Binomial transform is A045891. Partial sums are A008805. The sequence 0, 1, 0, 2, ... has a(n)=floor((n+2)/2)(1-(-1)^n)/2.

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

a(n) = (n+2)(n+3)/2 mod n+2. - Amarnath Murthy, Jun 17 2004

a(n) = (n+2)*(1 + (-1)^n)/4. - Bruno Berselli, Apr 01 2011

a(n) = A008619(n) * A059841(n). - Wesley Ivan Hurt, Jun 17 2013

E.g.f.: cosh(x) + x*sinh(x)/2. - Stefano Spezia, Mar 26 2022