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 strongly magic quad squares that can be made using the Quads-2^n deck. Here a strongly magic quad square is defined to be a 4-by-4 square of Quads cards so that if row/column numbers (0 to 3) of four cards form a quad, the cards have to form 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 for any four cells that have their row numbers (0 to 3), as well as column numbers (0 to 3), bitwise XOR to 0, the values in these cells have to bitwise XOR 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 (62,-1240,9920,-31744,32768).
FORMULA
a(n) = 2^n * (2^n - 1) * (2^n - 2) * (2^n - 4) * (2^n - 8).
a(n) = 322560 * A308436(n-4).
EXAMPLE
Such a square is uniquely defined by its first row and column. An example of such a square is 0,1,2,3/4,5,6,7/8,9,10,11/12,13,14,15.
MAPLE
a:= n-> mul(2^n-floor(2^i), i=-1..3):
seq(a(n), n=4..19); # Alois P. Heinz, May 24 2023
MATHEMATICA
Table[2^n (2^n - 1) (2^n - 2) (2^n - 4) (2^n - 8), {n, 4, 17}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and MIT PRIMES STEP senior group, May 07 2023
STATUS
approved