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%I #4 Mar 30 2012 17:34:22
%S 1,1,-1,-1,2,0,-1,-1,-1,3,4,-4,-7,1,5,0,-1
%N Hadamard 3 X 3 matrix substitution using 3 X 3 games matrices: MA = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; ( minimal Pisot matrix); MB = {{0, 0, 1}, {0, 1, 0},{1, 0, -1}}; With substitution rule: m[n]->If[m[n - 1][[i, j]] == 0, {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, If[m[n - 1][[i, j]] == 1, MA, MB]];.
%C Needs to be extended to the 27 X 27 level, but that is a long process by hand right now. This is the next Hadamard level up from the 2 X 2 -> 2^n to the 3 X 3 -> 3^n level matrices.
%F MA = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; MB = {{0, 0, 1}, {0, 1, 0},{1, 0, -1}}; With substitution rule: m[n]->If[m[n - 1][[i, j]] == 0, {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, If[m[n - 1][[i, j]] == 1, MA, MB]];
%e {1},
%e {1, -1},
%e {-1, 2, 0, -1},
%e {-1, -1, 3, 4, -4, -7, 1, 5, 0, -1}
%t MA = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; MB = {{0, 0, 1}, {0, 1, 0}, {1, 0, -1}}; m[0] = {{1}} m[1] = {{0, 0, 1}, {0, 1, 0}, {1, 0, -1}}; m[2] = {{0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 1, 0, 0, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1, 0, -1}}; m[n_] := Table[Table[If[m[n - 1][[i, j]] == 0, {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, If[m[n - 1][[i, j]] == 1, MA, MB]], {j, 1, 3^(n - 1)}], {i, 1, 3^(n - 1)}]; TableForm[m[3]] Table[CharacteristicPolynomial[m[i], x], {i, 0, 2}] a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[m[i], x], x], {i, 0, 2}]]; Flatten[a] (* visualization*) Table[ListDensityPlot[m[i]], {i, 0, 3}]
%K uned,sign
%O 1,5
%A _Roger L. Bagula_, Jan 25 2008
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