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%I #20 Jul 18 2023 07:36:11
%S 112,45280,4023232,136941952,3099135232,58520273920,1015268864512,
%T 16907404529152,275952876324352,4459246445032960,71702061084923392,
%U 1150074407046026752,18423955949551785472,294965554795552806400,4720907498205382415872,75546191122161343370752
%N Number of semimagic quads squares that can be formed using cards from Quads-2^n deck, where the first row and column are fixed.
%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 a(n) is the number of semimagic quads squares that can be made using the Quads-2^n deck (a generalization of the standard Quads-64 deck), where the first row and column are fixed. Here a semimagic quads square is defined as 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, and the first row and column are fixed.
%C Without loss of generality, the first row can be 0,1,2,3, and the first column 0,4,8,12.
%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_05">Index entries for linear recurrences with constant coefficients</a>, signature (31,-310,1240,-1984,1024).
%F a(n) = 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).
%F G.f.: -16*x^4*(7+2613*x+165892*x^2+1632480*x^3+2825728*x^4)/(x-1)/(4*x-1)/(2*x-1)/(8*x-1)/(16*x-1) . - _R. J. Mathar_, Jul 05 2023
%e An example of such a square is 0,1,2,3/4,5,6,7/8,9,10,11/12,13,14,15.
%p A362963 := proc(n)
%p 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) ;
%p end proc:
%p seq(A362963(n),n=4..24) ; # _R. J. Mathar_, Jul 05 2023
%t Table[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, 20}]
%Y Cf. A362874, A362964, A361495, A361613.
%K nonn,easy
%O 4,1
%A _Tanya Khovanova_ and MIT PRIMES STEP senior group, May 10 2023
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