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Search: id:A133756
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| A133756 |
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Based on the exceptional group sequence: a(n)=2*Gamma[n+3]+2^n where n=1+x/3 and the result is rounded to the nearest integer. A linearization of the dimensional results based on a least squares fit of the dimensions of the known exceptional groups. I use Round[] as giving a better result than Floor[]. This method is an approximation that relates to the idea of rational dimensions for sets. |
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| 5, 7, 10, 14, 21, 33, 52, 85, 144, 248, 438, 791, 1456, 2731, 5213, 10112, 19920, 39819, 80704, 165749, 344758, 725888, 1546398, 3331879
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The Linearization goes like this: a1 = {14, 52, 76, 133, 190, 248, 482}; (* dimensions of the exceptional groups known*) a2 = Table[ x /. FindRoot[2*Gamma[x + 3] + 2^x == a1[[n]], {x, 1 + Floor[n/3]}], {n, 1, 7}]; g1 = ListPlot[a2, PlotJoined -> True]; f[x_] = Fit[a2, {1, x}, x]; ( output:1.05097 + 0.348073 x) g2 = Plot[f[x], {x, 0, 7}]; Nearest rational line is: f(x)=1/x/3
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FORMULA
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f(n)=1+n/3 a(n) = Round[2*Gamma[f(n)+3]+2^f(n)]
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MATHEMATICA
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g[x_] = 1 + x/3; Table[Round[2*Gamma[g[x] + 3] + 2^g[x]], {x, -3, 20}]
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CROSSREFS
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Adjacent sequences: A133753 A133754 A133755 this_sequence A133757 A133758 A133759
Sequence in context: A070875 A091522 A020711 this_sequence A048584 A073895 A113194
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 01 2008
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