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 A134265 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. 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS Michelle Previte and Sean Yang, A Novel Way to Generate Fractals FORMULA 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}} EXAMPLE {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} MATHEMATICA 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}] CROSSREFS Cf. A122947, A131218. Sequence in context: A103921 A115623 A279044 * A182858 A175077 A001030 Adjacent sequences:  A134262 A134263 A134264 * A134266 A134267 A134268 KEYWORD tabf,uned,sign AUTHOR Roger L. Bagula, Jan 24 2008 STATUS approved

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Last modified September 22 17:05 EDT 2018. Contains 315270 sequences. (Running on oeis4.)