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Search: id:A123947
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| A123947 |
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A 5 X 5 vector matrix Markov where the M=M0.(I-M0)^(-1) matrix is from the 5 X 5 upper triangular Steinbach anti-diagonal matrix M0. Characteristic Polynomial is: -x - x^2 + 4 x^3 + 2 x^4 - x^5. |
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+0
1
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| 0, 1, 3, 9, 29, 90, 284, 890, 2797, 8780, 27574, 86581, 271881, 853732, 2680833, 8418132, 26433983, 83005929, 260648825, 818469251, 2570093890, 8070410030, 25342077544, 79577232067, 249882270390, 784660981474, 2463931734897
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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M / (I - M) = 0, -1, -1, 0, 1 -1, 0, 0, 0, -1 -1, 0, 1, 0, -1 0, 0, -1, 0, 0 1, -1, -1, 0, 1 = Q and using Q^n * [1, 0, 0, 0, 0] and taking the rightvector, I get a seq with a convergent which is one of the 11-gon diagonals. The polygon diagonals may be obtained through the formula: Sin (j + 1)Pi/N / Sin Pi/N and if j = 0 then we get 1, the edge. Thus the 11 gon diagonals are 1 1.918985868 2.6825069555 3.2287072825 3.5133369474 where our Q matrix has the convergent of ( Sin 4Pi/11) / ( Sin Pi/11.) The root structure is: {-1.27346, -0.437829, 0., 0.571167, 3.14012} The limiting ratio is: limit[a(n+1)/a(n),n->Infinity] = 3.14012
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REFERENCES
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Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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FORMULA
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M = {{0, -1, -1, 0, 1}, {-1, 0, 0, 0, -1}, {-1, 0, 1, 0, -1}, {0, 0, -1, 0, 0}, {1, -1, -1, 0, 1}}; v[1] = {0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = v[n][[1]]
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MATHEMATICA
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M = {{0, -1, -1, 0, 1}, {-1, 0, 0, 0, -1}, {-1, 0, 1, 0, -1}, {0, 0, -1, 0, 0}, {1, -1, -1, 0, 1}}; v[1] = {0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
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CROSSREFS
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Cf. A122773, A122771.
Adjacent sequences: A123944 A123945 A123946 this_sequence A123948 A123949 A123950
Sequence in context: A058145 A018361 A134325 this_sequence A135142 A098589 A024744
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KEYWORD
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nonn,uned,probation
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 26 2006
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