A361613 The number of magic quad squares that can be formed using cards from Quads-2^n deck.

3225600, 27398246400, 58912149381120, 35354354296504320, 13112764372566835200, 3994995853001760768000, 1112834567045660389539840, 296989774972633598731223040, 77616116494664898347650252800, 20075253208336550377420672204800, 5165728566421154658646365736796160

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

Table of n, a(n) for n=4..14.

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

Cf. A362874, A362963, A362964, A361495.

Sequence in context: A146976 A251529 A209715 * A235211 A219335 A203731

Adjacent sequences: A361610 A361611 A361612 * A361614 A361615 A361616

Keyword

nonn,easy

Author

Tanya Khovanova and MIT PRIMES STEP senior group, May 11 2023

Status

approved