A362963 Number of semimagic quads squares that can be formed using cards from Quads-2^n deck, where the first row and column are fixed.

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

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.

a(n) is the number of semimagic quads 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 semimagic quads square is defined as 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, 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

Table of n, a(n) for n=4..19.

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 (31,-310,1240,-1984,1024).

Formula

a(n) = 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).

G.f.: -16*x^4*(7+2613*x+165892*x^2+1632480*x^3+2825728*x^4)/(x-1)/(4*x-1)/(2*x-1)/(8*x-1)/(16*x-1) . - 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

A362963 := proc(n)
    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) ;
end proc:
seq(A362963(n), n=4..24) ; # R. J. Mathar, Jul 05 2023

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

Cf. A362874, A362964, A361495, A361613.

Sequence in context: A063409 A051366 A051363 * A304394 A210292 A270146

Adjacent sequences: A362960 A362961 A362962 * A362964 A362965 A362966

Keyword

nonn,easy

Author

Tanya Khovanova and MIT PRIMES STEP senior group, May 10 2023

Status

approved