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Number of 2 X 2 matrices with elements from {0,1,2,...,n} and with Nim-Determinant 1. (The Nim-Determinant of the 2 X 2 matrix [a,b; c,d] is defined to be a*d xor b*c, where * denotes Nim-Multiplication.)
0

%I #10 Oct 15 2022 17:31:56

%S 6,14,60,72,106,150,240,360,598,902,1308,1812,2378,3146,4080,4140,

%T 4270,4346,4500,4592,4754,4910,5144,5308,5558,5802,6140,6440,6826,

%U 7206,7680,8168,8702,9322,9996,10676,11418,12278,13208,14152,15158,16306,17540

%N Number of 2 X 2 matrices with elements from {0,1,2,...,n} and with Nim-Determinant 1. (The Nim-Determinant of the 2 X 2 matrix [a,b; c,d] is defined to be a*d xor b*c, where * denotes Nim-Multiplication.)

%C The six 2 X 2 matrices of 0's and 1's with Nim-Determinant 1, the 60 with elements 0,1,2 and 3 and the 4080 with elements from {0,1,...,15} form groups under Nim-Matrix-multiplication (use the corresponding Nim operation instead of ordinary multiplication and addition/subtraction).

%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, p. 443.

%D J. H. Conway, On Numbers and Games, chapter 6.

%K nonn

%O 1,1

%A _John W. Layman_, Mar 13 2001