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COMMENT
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Squareroot of M1 = A, the 8x8 matrix with the cyclotomic third roots of unity, mapped in a Gray code format. The 3 cyclotomic third roots of unity are (u, 1/u, and 1), where u = (-.5 + (sqrt(3)/2)i), = (1 Angle 120 deg.); and 1/u = (-.5, -(sqrt(3)/2)i), = (1 Angle -120 deg.). Third root = 1. Thus A^2 = M1. A =
[1, u, 1/u, u, 1/u, 1, 1/u, u;
u, 1, u, 1/u, 1, 1/u, u, 1/u;
1/u, u, 1, u, 1/u, u, 1/u, 1;
u, 1/u, u, 1, u, 1/u, 1, 1/u;
1/u, 1, 1/u, u, 1, u, 1/u, u;
1, 1/u, u, 1/u, u, 1, u, 1/u;
1/u, u, 1/u, 1, 1/u, u, 1, u;
u, 1/u, 1, 1/u, u, 1/u, u, 1]
A120096 = squares of A046717 terms; and is generated from analogous 4 X 4 matrices in a Gray code format.
A046717 can be generated from the analogous 2 X 2 matrix: P = [ -1,2; 2,-1 ], (which has the square root, [ u,1/u; 1/u;u ]). Then left term of P^n * [ 1,0 ], (unsigned) = 1, 5, 13, 41, 121...(where A046717 begins 1, 1, 5, 13...).
Pascal's triangle squared: (1; 2,1; 4,4,1; 8,12,6,1;...) rows can be generated by taking the dot product of the distinct terms (...4, 2, 1) in rows or columns of the analogous "M" matrices and their frequency: e.g. row 1 of the 8x8 matrix (unsigned) = [1, 2, 4, 2, 4, 8, 4, 2] with a frequency for (8, 4, 2, 1) being (1, 3, 3, 1). Dot product = the (8, 12, 6, 1) row of Pascal's Triangle squared.
Third powers of A046717: (deleting the first "1": (1, 5, 13, 41, 121, 365...)).
Leftmost term (unsigned) of M1^n * [1,0,0,0,0,0,0,0]; where M1 = 8x8 matrix:
[ -1, 2, -4, 2, -4, 8, -4, 2;
2, -1, 2, -4, 8, -4, 2, -4;
-4, 2, -1, 2, -4, 2, -4, 8;
2, -4, 2, -1, 2, -4, 8, -4;
-4, 8, -4, 2, -1, 2, -4, 2;
8, -4, 2, -4, 2, -1, 2, -4;
-4, 2, -4, 8, -4, 2, -1, 2;
2, -4, 8, -4, 2, -4, 2, -1]
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