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), where the first row and column are fixed. Here a magic quads square is defined as 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, 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 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 (15,-70,120,-64)
FORMULA
a(n) = 10 + 85*(2^n - 16) + 43*(2^n - 16)*(2^n - 32) + (2^n - 16)*(2^n - 32)*(2^n - 64).
G.f.: 2*x^4*(5+610*x+25144*x^2+101312*x^3)/ ( (x-1)*(8*x-1)*(2*x-1)*(4*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
A361495 := proc(n)
10 + 85*(2^n - 16) + 43*(2^n - 16)*(2^n - 32) + (2^n - 16)*(2^n - 32)*(2^n - 64)
end proc:
seq(A361495(n), n=4..20) ; # R. J. Mathar, Jul 13 2023
MATHEMATICA
Table[10 + 85(2^n - 16) + 43(2^n - 16)(2^n - 32) + (2^n - 16)(2^n - 32)(2^n - 64), {n, 4, 20}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and PRIMES STEP senior group, May 11 2023
STATUS
approved