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<a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
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R. H. Hardin, <a href="/A159359/b159359.txt">Table of n, a(n) for n = 2..100</a>
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Number of n X n arrays of squares of integers summing to 5.
12, 198, 4608, 53730, 378252, 1909236, 7628544, 25628076, 75297420, 198807114, 481029120, 1082267550, 2289691404, 4595197320, 8809614336, 16225724664, 28845544716, 49690719342, 83218759680, 135872231418, 216792905868, 338738351292, 519244496640, 782084374500
Empirical: n^2*(n^2-1)*(n^2+2)*(n^4-11*n^2+48)/120. [From _- _R. J. Mathar_, Aug 11 2009]
R. H. Hardin , Apr 11 2009
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As pointed out by Robert Israel in A159355, such arrangments of squares in an n X n array are related to the partitions of the sum (5 in this case). These partitions can be turned into a sum of products of binomial coefficients that computes the desired count, therefore all these sequences have holonomic recurrences. - Georg Fischer, Feb 17 2022
C:=binomial; seq(n^2*(n^2-1)+C(n^2, 5), n=2..22); # Georg Fischer, Feb 17 2022
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Number of nXn n X n arrays of squares of integers summing to 5