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Search: id:A134265
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| A134265 |
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Coefficients of the polynomials of a three level Hadamard matrix substitution set based on the game matrix set: MA={{0,1},{1,1}};MB={{1,0},{3,1}} Substitution rule is for m[n]:If[m[n - 1][[i, j]] == 0, {{0, 0}, {0, 0}}, If[m[n - 1][[i, j]] == 1, MA, MB]] Based on the Previte idea of graph substitutions as applied to matrices of graphs in the Fibonacci/ anti-Fibonacci game. |
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1
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| 1, 1, -1, 1, -2, 1, 1, 2, -1, -2, 1, 1, -2, -7, 6, 20, 6, -7, -2, 1, 1, 2, -25, -10, 225, -184, -498, 500, 610, -500, -498, 184, 225, 10, -25, -2, 1
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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m[n_] := Table[Table[If[m[n - 1][[i, j]] == 0, {{0, 0}, {0, 0}}, If[m[n - 1][[i, j]] == 1, ma, {{1, 0}, {3, 1}}]], {j, 1, 2^(n - 1)}], {i, 1, 2^(n - 1)}]
Michelle Previte and Sean Yang say Have you ever wanted to build your own fractal? This article will describe a procedure called a vertex replacement rule that can be used to construct fractals. We also show how one can easily compute the topological and box dimensions of the fractals resulting from vertex replacements.
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LINKS
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Michelle Previte and Sean Yang, A Novel Way to Generate Fractals
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FORMULA
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m[n] = If[m[n - 1][[i, j]] == 0, {{0, 0}, {0, 0}}, If[m[n - 1][[i, j]] == 1, MA, MB]] m[0] = {{1}} m[1] = {{1, 0}, {3, 1}} m[2] = {{0, 1, 0, 0}, {1, 1, 0, 0}, {1, 0, 0, 1}, {3, 1, 1, 1}} m[3] = {{0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 0}, {1, 1, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 1, 0, 1, 0, 1}, {3, 1, 1, 1, 1, 1, 1, 1}} m[4] = {{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1}, {1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}
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EXAMPLE
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{1},
{1, -1},
{1, -2, 1},
{1, 2, -1, -2, 1},
{1, -2, -7, 6, 20, 6, -7, -2,1},
{1, 2, -25, -10, 225, -184, -498, 500, 610, -500, -498,184, 225, 10, -25, -2, 1}
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MATHEMATICA
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m[0] = {{1}} m[1] = {{1, 0}, {3, 1}} m[2] = {{0, 1, 0, 0}, {1, 1, 0, 0}, {1, 0, 0, 1}, {3, 1, 1, 1}} m[3] = {{0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 0}, {1, 1, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 1, 0, 1, 0, 1}, {3, 1, 1, 1, 1, 1, 1, 1}} m[4] = {{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1}, {1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}; Table[CharacteristicPolynomial[m[i], x], {i, 0, 4}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[m[i], x], x], {i, 0, 4}]]; Flatten[a] (* visualization*) Table[ListDensityPlot[m[i]], {i, 0, 4}]
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CROSSREFS
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Cf. A122947, A131218.
Adjacent sequences: A134262 A134263 A134264 this_sequence A134266 A134267 A134268
Sequence in context: A007421 A103921 A115623 this_sequence A001030 A071709 A131406
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KEYWORD
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tabf,uned,probation,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 24 2008
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