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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. 5
112, 45280, 4023232, 136941952, 3099135232, 58520273920, 1015268864512, 16907404529152, 275952876324352, 4459246445032960, 71702061084923392, 1150074407046026752, 18423955949551785472, 294965554795552806400, 4720907498205382415872, 75546191122161343370752 (list; graph; refs; listen; history; text; internal format)
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
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.
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
Sequence in context: A063409 A051366 A051363 * A304394 A210292 A270146
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and MIT PRIMES STEP senior group, May 10 2023
STATUS
approved

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Last modified June 13 05:35 EDT 2024. Contains 373366 sequences. (Running on oeis4.)