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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.

%I A154232 #17 Mar 03 2021 09:45:53
%S A154232 0,1,-1,-1,0,-1,5,-5,16,-55,35,-1045,64,-30305,105,-1242505,160,
%T A154232 -68337775,231,-4851982025,320,-431826400225,429,-47069077624525,560,
%U A154232 -6166049168812775,715,-955737621165980125,896,-172988509431042402625
%N 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.
%C A154232 Essentially A077415 and A130031 interleaved, see formulas.
%H A154232 G. C. Greubel, <a href="/A154232/b154232.txt">Table of n, a(n) for n = 0..351</a>
%F A154232 From _Robert Israel_, Sep 06 2016: (Start)
%F A154232 a(2*n) = A077415(n) for n >= 2.
%F A154232 a(2*n+1) = cos(Pi*sqrt(5)/2)*Gamma(n+1/2-sqrt(5)/2)*Gamma(n+1/2+sqrt(5)/2)/Pi.
%F A154232 a(2*n+1) = (-1)^n*A130031(n). (End)
%p A154232 a[0]:= 0: a[1]:= 1:
%p A154232 for n from 1 to 49 do
%p A154232   a[2*n]:= (n^2-n-1) +a[2*n-2];
%p A154232   a[2*n+1]:= (n^2-n-1)*a[2*n-1];
%p A154232 od:
%p A154232 seq(a[i],i=0..99); # _Robert Israel_, Sep 06 2016
%t A154232 (* First program *)
%t A154232 b[n_]:= b[n]= If[n==0, 0, n^2 -n -1 + b[n-1]];
%t A154232 c[n_]:= c[n]= If[n==0, 1, (n^2 -n -1)*c[n-1]];
%t A154232 Flatten[Table[{b[n], c[n]}, {n, 0, 15}]] (* modified by _G. C. Greubel_, Mar 02 2021 *)
%t A154232 (* Second program *)
%t A154232 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]]];
%t A154232 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Mar 02 2021 *)
%o A154232 (Sage)
%o A154232 def a(n):
%o A154232     if (n<2): return n
%o A154232     elif (n%2==0): return ((n^2-2*n-4)/4) + a(n-2)
%o A154232     else: return ((n^2-4*n-1)/4)*a(n-2)
%o A154232 [a(n) for n in (0..40)] # _G. C. Greubel_, Mar 02 2021
%o A154232 (Magma)
%o A154232 function a(n)
%o A154232   if n lt 2 then return n;
%o A154232   elif (n mod 2 eq 0) then return ((n^2-2*n-4)/4) + a(n-2);
%o A154232   else return ((n^2-4*n-1)/4)*a(n-2);
%o A154232   end if; return a;
%o A154232 end function;
%o A154232 [a(n): n in [0..40]]; // _G. C. Greubel_, Mar 02 2021
%Y A154232 Cf. A077415, A130031.
%K A154232 sign
%O A154232 0,7
%A A154232 _Roger L. Bagula_, Jan 05 2009
%E A154232 Edited by _Robert Israel_, Sep 06 2016