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A120719 Second level Hadamard-Sylvester matrix self-similarity for the 2 X 2 Fibonacci matrix as a 16 X 16 matrix Markov ( made using an array repartitioning method) Characteristic Polynomial:1 - x - 40 x^2 - 45 x^3 + 285 x^4 + 272 x^5 - 1022 x^6 - 370 x^7 + 1840 x^8 - 370 x^9 - 1022 x^10 + 272 x^11 + 285 x^12 - 45 x^13 - 40x^14 - x^15 + x^16. 0
0, 610, 1596, 16500, 97410, 707560, 4744080, 32791746, 224035980, 1537454500, 10532923170, 72206679000, 494878036896, 3392033285410, 23249109634140, 159352376426580, 1092215843858370, 7486162932788296, 51310913160533040 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

With the right starting matrix Pascal's triangle is obtained by repeated Haramard -Silvester Matrix Self-Similar operations.

LINKS

Table of n, a(n) for n=1..19.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

FORMULA

M={{0, 0, 0, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1}, {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1}, {0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1}, {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1}, {0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}} v[1] = {0, 1, 2, 3} v[n_] := v[n] = M.v[n - 1] a(n) = v[n][[1]]

G.f.: -2*x^2*(60*x^3-315*x^2-727*x+305)/((x-1)*(x^2-7*x+1)*(x^2+3*x+1)). [Colin Barker, Nov 01 2012]

MATHEMATICA

t[n_, m_] := If[ n == m == 1, 0, 1] a = Table[t[n, m]*t[i, j], {n, 1, 2}, {m, 1, 2}, {i, 1, 2}, {j, 1, 2}]; M = Flatten[Table[{Flatten[Table[a[[ n, m]][[1, i]], {n, 1, 2}, {i, 1, 2}]], Flatten[Table[a[[n, m]][[2, i]], {n, 1, 2}, {i, 1, 2}]]}, {m, 1, 2}], 1] aa = Table[M[[n, m]]*M[[i, j]], {n, 1, 4}, {m, 1, 4}, {i, 1, 4}, {j, 1, 4}]; M2 = Flatten[Table[{Flatten[Table[aa[[ n, m]][[1, i]], {n, 1, 4}, {i, 1, 4}]], Flatten[Table[aa[[n, m]][[2, i]], {n, 1, 4}, {i, 1, 4}]], Flatten[Table[aa[[ n, m]][[3, i]], {n, 1, 4}, {i, 1, 4}]], Flatten[Table[aa[[ n, m]][[4, i]], {n, 1, 4}, {i, 1, 4}]]}, {m, 1, 4}], 1] v[1] = Table[Fibonacci[n], {n, 0, 15}] v[n_] := v[n] = M2.v[n - 1] a = Table[v[n][[1]], {n, 1, 50}] Det[M2 - x*IdentityMatrix[16]] Factor[%] aaa = Table[x /. NSolve[Det[M2 - x*IdentityMatrix[16]] == 0, x][[n]], {n, 1, 16}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]

CROSSREFS

Cf. A000045, A072845.

Sequence in context: A301561 A204487 A090177 * A045730 A072317 A220568

Adjacent sequences:  A120716 A120717 A120718 * A120720 A120721 A120722

KEYWORD

nonn,easy,uned

AUTHOR

Roger L. Bagula, Aug 13 2006

STATUS

approved

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Last modified December 3 21:17 EST 2021. Contains 349468 sequences. (Running on oeis4.)