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 magic quad squares that can be made using the Quads-2^n deck (a generalization of the standard Quads-64 deck). Here a magic quad square is defined to be a 4-by-4 square of Quads cards so that each row, column, and diagonal 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, column, and diagonal bitwise XORs to 0.
LINKS
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 (510,-86360,6217920,-205605888,3183575040,-22638755840,68451041280,-68719476736).
FORMULA
a(n) = 2^n(2^n - 1)(2^n - 2)(2^n - 4)(2^n - 8)(10 + 85(2^n - 16) + 43(2^n - 16)(2^n - 32) + (2^n - 16)(2^n - 32)(2^n - 64)).
G.f.: 645120*x^4*(5+39920*x+70091776*x^2+11866341376*x^3) / ( (4*x-1) *(256*x-1) *(64*x-1) *(2*x-1) *(8*x-1) *(128*x-1) *(16*x-1) *(32*x-1) ). - R. J. Mathar, Jul 13 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
A361613 := proc(n)
2^n*(2^n - 1)*(2^n - 2)*(2^n - 4)*(2^n - 8)*(10 + 85*(2^n - 16) + 43*(2^n - 16)*(2^n - 32) + (2^n - 16)*(2^n - 32)*(2^n - 64))
end proc:
seq(A361613(n), n=4..30) ; # R. J. Mathar, Jul 13 2023
MATHEMATICA
Table[2^n (2^n - 1) (2^n - 2) (2^n - 4) (2^n - 8) (10 + 85 (2^n - 16) + 43 (2^n - 16) (2^n - 32) + (2^n - 16) (2^n - 32) (2^n - 64)), {n, 4, 15}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and MIT PRIMES STEP senior group, May 11 2023
STATUS
approved