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%I #17 Feb 08 2014 17:28:42
%S 1,-20,140,200,-67600,3983200,-228488000,14375920000,-1002261520000,
%T 74864404160000,-5398716356800000,221997813232000000,
%U 54286859023072000000,-27326116497867200000000,9481971502321385600000000,-3155347494162485190400000000
%N Real part of the product of all the Gaussian integers in the rectangle [1, 1] to [2, n].
%C By Gaussian integers, we mean complex numbers of the form a + bi, where both a and b are integers in Z, i = sqrt(-1). Thus the quadratic integer ring under consideration here is Z[i].
%F a(n) +(2*n+3)*(n-2)*a(n-1) +n*(n+1)*(n^2-4*n+8)*a(n-2) -2*(n^2-4*n+8)*(n^2-4*n+5)*a(n-3)=0. - _R. J. Mathar_, Feb 08 2014
%e For n = 2, we have (1 + i)(1 + 2i)(2 + i)(2 + 2i) which gives -20 + 0i, so a(2) = -20.
%t Table[Re[Times@@Flatten[Table[a + b I, {a, 2}, {b, n}]]], {n, 20}] (* _Alonso del Arte_, Feb 02 2014 *)
%o (JavaScript)
%o function cNumber(x, y) {
%o return [x, y];
%o }
%o function cMult(a, b) {
%o return [a[0] * b[0] - a[1] * b[1], a[0] * b[1] + a[1] * b[0]];
%o }
%o for (i = 1; i < 20; i++) {
%o c = cNumber(1, 0);
%o for (j = 1; j <= 2; j++)
%o for (k = 1; k <= i; k++)
%o c = cMult(c, cNumber(j, k));
%o document.write(c[0] + ", ");
%o }
%o (PARI) a(n) = real(prod(i=1, 2, prod(j=1, n, i+I*j))); \\ _Michel Marcus_, Feb 03 2014
%Y Cf. A105750, A234459, A204041.
%K sign
%O 1,2
%A _Jon Perry_, Feb 02 2014
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