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First Hadamard-Sylvester matrix self -similar matrix based on the Padovan/ Minimal Pisot 3 X 3 matrix as an 9 X 9 matrix: Characteristic Polynomial:1 - x - x^3 - x^4 - x^5 + 3 x^6 + 2 x^7 - x^9.
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%I #11 Jul 28 2015 15:33:09

%S 0,13,5,22,42,54,126,192,347,631,1056,1914,3320,5814,10276,17921,

%T 31549,55338,97026,170454,298914,524684,920815,1615647,2835660,

%U 4975898,8732160,15324202,26891432,47191909,82815621,145331022,255039162

%N First Hadamard-Sylvester matrix self -similar matrix based on the Padovan/ Minimal Pisot 3 X 3 matrix as an 9 X 9 matrix: Characteristic Polynomial:1 - x - x^3 - x^4 - x^5 + 3 x^6 + 2 x^7 - x^9.

%C As far as I can tell by searching the Internet, this matrix and this approach to sequences is entirely new and unique. The second of these matrices at 81 X 81 gives a new fractal that is Cantor dust like. aa = Table[M[[n, m]]*M[[i, j]], {n, 1, 9 }, {m, 1, 9}, {i, 1, 9}, {j, 1, 9}]; M2 = Flatten[Table[{Flatten[Table[aa[[ n, m]][[1, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[n, m]][[2, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[3, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[4, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[5, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[n, m]][[6, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[7, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[8, i]], {n, 1, 9}, {i, 1, 9}]], Flatten[Table[aa[[ n, m]][[9, i]], {n, 1, 9}, {i, 1, 9}]]}, {m, 1, 9}], 1]; ListDensityPlot[M2, Mesh -> False]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, -1, 1, -1).

%F M={{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, 1, 0, 0, 0, 0, 0, 1, 0}, { 0, 0, 1, 0, 0, 0, 0, 0, 1}, {1, 1, 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}} v[1] = Table[Fibonacci[n], {n, 0, 8}] v[n_] := v[n] = M.v[n - 1] a(n) = v[n][[1]]

%F G.f.: x^2(13-8x+4x^2+2x^3-2x^4)/((1-2x+x^2-x^3)(1+x-x^3)). a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-4) +a(n-5) -a(n-6). Partial fraction decomposition yield decomposition in terms of A005314 and A050935. [From _R. J. Mathar_, Nov 26 2008]

%t Clear[t, M, a, v, a0] t[n_, m_] := {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}[[n, m]] a0 = Table[t[n, m]*t[i, j], {n, 1, 3}, {m, 1, 3}, {i, 1, 3}, {j, 1, 3}]; M = Flatten[Table[{Flatten[Table[a0[[ n, m]][[1, i]], {n, 1, 3}, {i, 1, 3}]], Flatten[Table[a0[[n, m]][[2, i]], {n, 1, 3}, {i, 1, 3}]], Flatten[Table[a0[[n, m]][[3, i]], {n, 1, 3}, {i, 1, 3}]]}, {m, 1, 3}], 1] v[1] = Table[Fibonacci[n], {n, 0, 8}] v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}] Det[M - x*IdentityMatrix[9]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[9]] == 0, x][[n]], {n, 1, 9}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}] ListDensityPlot[M, Mesh -> False]

%Y Cf. A000931.

%K nonn,uned,obsc

%O 1,2

%A _Roger L. Bagula_, Aug 13 2006