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 half-half.
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.
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.
LINKS
Ray Chandler, Table of n, a(n) for n = 4..100
Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, and Raphael Walker, How many cards should you lay out in a game of EvenQuads? A detailed study of 2-caps in AG(n,2), arXiv:2212.05353 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1022, -347480, 50434240, -3389180928, 108453789696, -1652629176320, 11659494031360, -35115652612096, 35184372088832).
FORMULA
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)).
EXAMPLE
An example of a such square is 0,1,2,3/4,5,6,7/8,9,10,11/12,13,14,15.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Tanya Khovanova and MIT PRIMES STEP senior group, May 10 2023
STATUS
approved