%I #21 Oct 12 2012 14:00:57
%S 1,179,204,108,996,2934,81720,2152080,851040,271106640,7935621120,
%T 1143137318400,52250931532800
%N Numbers n such that s! + n^2 and (s + 2)! + n^2 are squares for some s, ordered by s.
%C The corresponding numbers s are: 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 19, 25, 27.
%e 5! + 1 = 11^2 and 7! + 1 = 71^2.
%e 6! + 179^2 = 181^2 and 8! + 179^2 = 269^2.
%e 7! + 204^2 = 216^2 and 9! + 204^2 = 636^2.
%e 8! + 108^2 = 228^2 and 10! + 108^2 = 1908^2.
%o (PARI) for(n=4,32,a=n!;b=((n+2)*(n+1)-1)*a;c=divisors(b);for(i=2,#c-1,s=c[i];r=b\s;if(r<s,next(2),d=abs(s-r)/2;t=d^2-a;if(issquare(t),print([n,d,sqrtint(t)])))))
%Y Cf. A217277, A217541, A217550, A217551, A217553.
%K nonn
%O 1,2
%A _Robin Garcia_, Oct 06 2012