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Search: id:A147869
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| A147869 |
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A Fernandez-type expansion for Conway's A004001: p(x,n)=Product[x+A004001[n],{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)). |
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2
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| 1, 1, 1, 3, 4, 7, 11, 17, 25, 41, 59, 86, 125, 180, 263, 382, 536, 738, 1073, 1466, 2028, 2841, 3889, 5275, 7211, 9800, 13249, 17860, 23948, 31921, 42864, 56802, 75115, 99788, 131239, 172870, 226789, 296404, 386745, 504939, 655227, 849628, 1101270
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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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:1.2125507033495728 or smaller: Table[N[a2[[n + 1]]/a2[[n]]], {n, 1, 99}]. This result is interesting, important and new, since nothing except Salems have that low of a ratio.
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FORMULA
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p(x,n)=Product[x+A004001[n],{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)).
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MATHEMATICA
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(*A004001*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]]; P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}]; a2 = CoefficientList[ExpandAll[P[x, 100]], x]; Table[a2[[n]], {n, 1, 100}]
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CROSSREFS
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A147559, A147654, A147655, A004001
Sequence in context: A166375 A060962 A069950 this_sequence A100581 A093090 A000204
Adjacent sequences: A147866 A147867 A147868 this_sequence A147870 A147871 A147872
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KEYWORD
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nonn
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 16 2008
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