%I #23 Aug 09 2023 18:19:36
%S 36126720,905542041600,3379298591047680,4173723561555394560,
%T 3211490275093527920640,1999224826411677961420800,
%U 1126422803027648324787240960,604702030645626380104520171520,316988224809023889673442568437760
%N Number of semimagic quad squares that can be formed using cards from Quads-2^n deck.
%C This sequence is related to the game of EvenQuads: a deck of 64 cards with 3 attributes and 4 values in each attribute. Four cards form a quad when for every attribute, the values are either the same, all different, or half-half.
%C This sequence counts the semimagic quad squares that can be made using the Quads-2^n deck (a generalization of the standard Quads-64 deck). Here a semimagic quad square is defined to be a 4-by-4 square of Quads cards so that each row and column forms a quad.
%C a(n) is the number of 4-by-4 squares that can be made out of distinct numbers in the range from 0 to 2^n-1, so that each row and column bitwise XORs to 0.
%H Ray Chandler, <a href="/A362964/b362964.txt">Table of n, a(n) for n = 4..100</a>
%H Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, and Raphael Walker, <a href="https://arxiv.org/abs/2212.05353">How many cards should you lay out in a game of EvenQuads? A detailed study of 2-caps in AG(n,2)</a>, arXiv:2212.05353 [math.CO], 2023.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1022, -347480, 50434240, -3389180928, 108453789696, -1652629176320, 11659494031360, -35115652612096, 35184372088832).
%F a(n) = 2^n * (2^n - 1) * (2^n - 2) * (2^n - 4) * (2^n - 8) * (112 + 2823 * (2^n - 16) + 2531 * (2^n - 16) * (2^n - 32) + 159 * (2^n - 16) * (2^n - 32) * (2^n - 64) + (2^n - 16) * (2^n - 32) * (2^n - 64) * (2^n - 128)).
%e An example of a such square is 0,1,2,3/4,5,6,7/8,9,10,11/12,13,14,15.
%t Table[2^n (2^n - 1) (2^n - 2) (2^n - 4) (2^n - 8) (112 + 2823 (2^n - 16) + 2531 (2^n - 16) (2^n - 32) + 159 (2^n - 16) (2^n - 32) (2^n - 64) + (2^n - 16) (2^n - 32) (2^n - 64) (2^n - 128)), {n, 4, 15}]
%Y Cf. A362874, A362963, A361495, A361613.
%K nonn
%O 4,1
%A _Tanya Khovanova_ and MIT PRIMES STEP senior group, May 10 2023