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A134325
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A 9 X 9 Matrix vector sum Markov sequence with characteristic polynomial: -1 - 7 x + 43 x^2 + 48 x^3 - 38 x^4 - 47 x^5 + 7 x^6 + 13 x^7 - x^9 Largest root/ratio is 3.14065< Pi such that : a(n)/Pi^(n+1)->fixed number C less than one ( about 0.885174 ) a(n)~C*Pi^(n+1).
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0
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3, 9, 29, 87, 281, 857, 2741, 8471, 26876, 83710, 264309, 826648, 2603282, 8159120, 25659956, 80507046, 253016149, 794239479, 2495264294, 7834844821, 24610555850, 77283962750, 242741417095, 762321637062, 2394279296957
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| The root structure is nearly all semi-unique algebriac irrationals: NSolve[CharacteristicPolynomial[M, x] == 0, x] {{x -> -2.1893697030273365`}, { x -> -1.8207023936551185`}, {x -> -1.2128832881490639` - 0.35413663859184796`I}, {x -> -1.2128832881490639` + 0.35413663859184796` I}, {x -> -0.09433091961414544`}, {x -> 0.223526912678973`}, {x -> 1.2182640154457587`}, {x -> 1.9477296343029344`}, {x -> 3.140649030167062`}} The game value of the matrix is: Det[M]/(Sum[Sum[If[i == j, M[[i, j]], 0], {i, 1, 9}], {j, 1, 9}] - Sum[Sum[If[i == j, 0, M[[i, j]]], {i, 1, 9}], {j, 1, 9}]) 1/(3^3+1)=1/28
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FORMULA
| M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 0, -1}}; v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = Sum[v[n][[i]],{i,1,9}]
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MATHEMATICA
| M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 0, -1}}; v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 0, 50}]
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CROSSREFS
| Sequence in context: A161590 A192245 A018361 * A123947 A135142 A195624
Adjacent sequences: A134322 A134323 A134324 * A134326 A134327 A134328
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 16 2008
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