Padovan Minimal Pisot Binet solution in Mathematica
- From: Roger Bagula <rlbagulatftn@xxxxxxxxx>
- Date: Mon, 27 Feb 2006 16:31:04 GMT
This Minimal Pisot is also a fractal tile generator of some note!
Compare the two outputs:
{0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616}
{-1, 1, 0, 1, 2, 2, 3, 4, 4, 7, 8, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465}
Notebook:
M = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}
w[0] = {0, 1, 1};
w[n_] := w[n] = M.w[n - 1]
a0 = Table[w[n][[1]], {n, 0, 25}]
Det[M - x*IdentityMatrix[3]]
NSolve[Det[M - x*IdentityMatrix[3]] == 0, x]
NSolve[x^3 - x - 1 == 0, x]
b0 = 1.32471795724474605`;
b1 = -0.662358978622373051` - 0.562279512062301289` I;
b2 = -0.662358978622373051` + 0.562279512062301289` I;
g[n_] := N[(x*b0^n + y*b1^n + z*b2^n)];
digits = 25
a = Table[g[n], {n, 1, digits}]
Solve[Table[a[[n]] - a0[[n]] == 0, {n, 1, 3}], {x, y, z}]
Solve[Table[a[[n]] - a0[[n]] == 0, {n, 2, 4}], {x, y, z}]
Solve[Table[a[[n]] - a0[[n]] == 0, {n, 3, 5}], {x, y, z}]
x = 0.4114955886626457;
y = 0.29425220566867694- 0.13811139432288674*I;
z = 0.29425220566867716+ 0.13811139432288672*I;
a2 = Table[Floor[Re[g[n]]], {n, 1, 25}]
Compare the two outputs:
{0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616}
{-1, 1, 0, 1, 2, 2, 3, 4, 4, 7, 8, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465}
Roger L. Bagula { email: rlbagula@xxxxxxxxxxxxx or rlbagulatftn@xxxxxxxxx }
11759 Waterhill Road,
Lakeside, Ca. 92040 telephone: 619-561-0814
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