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A147871
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A Fernandez-type expansion for A147665: p(x,n)=Product[x+A147665[n],{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)).
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1
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1, 1, 1, 3, 4, 7, 10, 15, 24, 37, 49, 73, 105, 142, 208, 294, 391, 538, 752, 988, 1359, 1812, 2410, 3232, 4270, 5598, 7454, 9721, 12639, 16625, 21445, 27649, 35793, 46235, 59141, 76215, 96975, 123262, 157671, 199625, 252591, 319792, 403262, 507682
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The resulting polynomial is equivalent to and 1/q(x) type expansion of q(x)=x^n*f(1/x) and that expansion has a limiting ratio ( largest positive root) near:1.1847680405611645 or smaller: Table[N[a2[[n + 1]]/a2[[n]]], {n, 1, 99}]. This result is interesting, important and new, since it is lower than the result for Conway's A004001 sequence.
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FORMULA
| p(x,n)=Product[x+A147665[n],{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)).
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MATHEMATICA
| Clear[f, P, n]; (*A147665*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + If[Mod[ n, 3] == 0, f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]]; P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}]; Length[CoefficientList[ExpandAll[P[x, 99]], x]]; a1 = CoefficientList[ExpandAll[P[x, 99]], x]; a2 = CoefficientList[ExpandAll[P[x, 100]], x]; a = Sum[If[a1[[n]] - a2[[n]] == 0, 1, 0], {n, 1, 4951}]; Table[a2[[n]], {n, 1, 100}]
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CROSSREFS
| A147559, A147654, A147655, A004001, A147665
Sequence in context: A105343 A147789 A047625 * A004397 A047967 A147955
Adjacent sequences: A147868 A147869 A147870 * A147872 A147873 A147874
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 16 2008
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