A134655 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]];.

1, 1, -1, -1, 2, 0, -1, -1, -1, 3, 4, -4, -7, 1, 5, 0, -1

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..17.

Formula

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]];

Example

{1},
{1, -1},
{-1, 2, 0, -1},
{-1, -1, 3, 4, -4, -7, 1, 5, 0, -1}

Mathematica

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}]

Crossrefs

Sequence in context: A024996 A187596 A263863 * A262124 A199954 A333580

Adjacent sequences: A134652 A134653 A134654 * A134656 A134657 A134658

Keyword

uned,sign

Author

Roger L. Bagula, Jan 25 2008

Status

approved