A154232 a(2n) = (n^2-n-1) + a(2n-2), a(2n+1) = (n^2-n-1)*a(2n-1), with a(0)=0 and a(1)=1.

0, 1, -1, -1, 0, -1, 5, -5, 16, -55, 35, -1045, 64, -30305, 105, -1242505, 160, -68337775, 231, -4851982025, 320, -431826400225, 429, -47069077624525, 560, -6166049168812775, 715, -955737621165980125, 896, -172988509431042402625

Offset: 0

Author

Roger L. Bagula, Jan 05 2009

Keywords

• sign

Comments

Essentially A077415 and A130031 interleaved, see formulas.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..351

Crossrefs

Cf. A077415, A130031.

Programs

Magma

function a(n)
  if n lt 2 then return n;
  elif (n mod 2 eq 0) then return ((n^2-2*n-4)/4) + a(n-2);
  else return ((n^2-4*n-1)/4)*a(n-2);
  end if; return a;
end function;
[a(n): n in [0..40]]; // G. C. Greubel, Mar 02 2021

Maple

a[0]:= 0: a[1]:= 1:
for n from 1 to 49 do
  a[2*n]:= (n^2-n-1) +a[2*n-2];
  a[2*n+1]:= (n^2-n-1)*a[2*n-1];
od:
seq(a[i],i=0..99); # Robert Israel, Sep 06 2016

Mathematica

(* First program *)
b[n_]:= b[n]= If[n==0, 0, n^2 -n -1 + b[n-1]];
c[n_]:= c[n]= If[n==0, 1, (n^2 -n -1)*c[n-1]];
Flatten[Table[{b[n], c[n]}, {n, 0, 15}]] (* modified by G. C. Greubel, Mar 02 2021 *)
(* Second program *)
a[n_]:= a[n]= If[n<2, n, If[EvenQ[n], ((n^2-2*n-4)/4) + a[n-2], ((n^2-4*n-1)/4)*a[n-2]]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Mar 02 2021 *)

Sage

def a(n):
    if (n<2): return n
    elif (n%2==0): return ((n^2-2*n-4)/4) + a(n-2)
    else: return ((n^2-4*n-1)/4)*a(n-2)
[a(n) for n in (0..40)] # G. C. Greubel, Mar 02 2021

Formula

From Robert Israel, Sep 06 2016: (Start)

a(2*n) = A077415(n) for n >= 2.

a(2*n+1) = cos(Pi*sqrt(5)/2)*Gamma(n+1/2-sqrt(5)/2)*Gamma(n+1/2+sqrt(5)/2)/Pi.

a(2*n+1) = (-1)^n*A130031(n). (End)

Extensions

Edited by Robert Israel, Sep 06 2016