%I #22 Aug 09 2023 18:19:04

%S 10,1370,70138,1159994,12654010,116939450,1003021498,8303802554,

%T 67568410810,545138438330,4379550748858,35110336483514,

%U 281178729140410,2250614613070010,18009657286316218,144096222341746874,1152845639987482810

%N Number of magic quad 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 This sequence counts the magic quad 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 magic quads square is defined as a 4-by-4 square of Quads cards so that each row, column, and diagonal 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, column, and diagonal 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64)

%F a(n) = 10 + 85*(2^n - 16) + 43*(2^n - 16)*(2^n - 32) + (2^n - 16)*(2^n - 32)*(2^n - 64).

%F G.f.: 2*x^4*(5+610*x+25144*x^2+101312*x^3)/ ( (x-1)*(8*x-1)*(2*x-1)*(4*x-1) ). - _R. J. Mathar_, Jul 13 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 A361495 := proc(n)

%p 10 + 85*(2^n - 16) + 43*(2^n - 16)*(2^n - 32) + (2^n - 16)*(2^n - 32)*(2^n - 64)

%p end proc:

%p seq(A361495(n),n=4..20) ; # _R. J. Mathar_, Jul 13 2023

%t Table[10 + 85(2^n - 16) + 43(2^n - 16)(2^n - 32) + (2^n - 16)(2^n - 32)(2^n - 64), {n, 4, 20}]

%Y Cf. A362874, A362963, A362964, A361613.

%K nonn,easy

%O 4,1

%A _Tanya Khovanova_ and PRIMES STEP senior group, May 11 2023