A121230 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.

0, 13, 5, 22, 42, 54, 126, 192, 347, 631, 1056, 1914, 3320, 5814, 10276, 17921, 31549, 55338, 97026, 170454, 298914, 524684, 920815, 1615647, 2835660, 4975898, 8732160, 15324202, 26891432, 47191909, 82815621, 145331022, 255039162

Offset: 1

Author

Roger L. Bagula, Aug 13 2006

Keywords

• nonn
• uned
• obsc

Comments

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]

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, -1, 1, -1).

Crossrefs

Cf. A000931.

Programs

Mathematica

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]

Formula

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]]

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]