

A362963


Number of semimagic quads squares that can be formed using cards from Quads2^n deck, where the first row and column are fixed.


5



112, 45280, 4023232, 136941952, 3099135232, 58520273920, 1015268864512, 16907404529152, 275952876324352, 4459246445032960, 71702061084923392, 1150074407046026752, 18423955949551785472, 294965554795552806400, 4720907498205382415872, 75546191122161343370752
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OFFSET

4,1


COMMENTS

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 halfhalf.
a(n) is the number of semimagic quads squares that can be made using the Quads2^n deck (a generalization of the standard Quads64 deck), where the first row and column are fixed. Here a semimagic quads square is defined as a 4by4 square of Quads cards so that each row and column forms a quad.
a(n) is the number of 4by4 squares that can be made out of distinct numbers in the range from 0 to 2^n1, so that each row and column bitwise XORs to 0, and the first row and column are fixed.
Without loss of generality, the first row can be 0,1,2,3, and the first column 0,4,8,12.


LINKS

Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel RoseLevine, Darrion Thornburgh, and Raphael Walker, How many cards should you lay out in a game of EvenQuads? A detailed study of 2caps in AG(n,2), arXiv:2212.05353 [math.CO], 2023.


FORMULA

a(n) = 112 + 2823*(2^n16) + 2531*(2^n16)*(2^n32) + 159*(2^n16)*(2^n32)*(2^n64) + (2^n16)*(2^n32)*(2^n64)*(2^n128).
G.f.: 16*x^4*(7+2613*x+165892*x^2+1632480*x^3+2825728*x^4)/(x1)/(4*x1)/(2*x1)/(8*x1)/(16*x1) .  R. J. Mathar, Jul 05 2023


EXAMPLE

An example of such a square is 0,1,2,3/4,5,6,7/8,9,10,11/12,13,14,15.


MAPLE

112 + 2823*(2^n16) + 2531*(2^n16)*(2^n32) + 159*(2^n16) *(2^n32) *(2^n64) + (2^n16) *(2^n32) *(2^n64)*(2^n128) ;
end proc:


MATHEMATICA

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}]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



