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Search: id:A113976
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| A113976 |
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Farey: coefficient expansion of a cubic over cubic that has 123 roots and a Farey p[1/2]=1. |
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| 14, 133, 616, 2128, 6664, 20272, 61096, 183568, 550984, 1653232, 4959976, 14880208, 44640904, 133922992, 401769256, 1205308048, 3615924424, 10847773552, 32543320936, 97629963088, 292889889544, 878669668912
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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I was trying to get an integer root cubic over cubic that had the Farey conditions: p[1/2]=1;p[0]=0;p[1]=0
The function has the characteristic Farey shape: fa[x_] := 1/p[x] /; 0 <= x <= 1/2 fa[x_] := p[x] /; 1/2 < x <= 1
My theory was that there should be a family of Farey type functions of the form: p[x]=k*(x - a)*(x - b)*(x -c)/((x - d)*(x - e)*(x - f)) with integer root ( rational) structure ( simply starting at abc=123): Abs[a*b*c]=Abs[d*f] and sign[a*b*c]=Sign[d*f]; e=0
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FORMULA
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b(n)= coefficient expansion of (-7/5)*(x-1)*(x-2)*(x-3)/(x*(x+3)*(x+1) a(n) = b(n)
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MATHEMATICA
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a = 1; b = 2; c = 3; d = -3; e = 0; f = -1; p[x_] = FullSimplify[ExpandAll[(-7/5)*(x - a)*(x - b)*(x -c)/((x - d)*(x - e)*(x - f))]] a = Abs[ReplacePart[Table[5*Coefficient[Series[p[x], {x, 0, 30}], x^n]*3^(n + 1), {n, -1, 30}], -133, 2]]
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CROSSREFS
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Cf. A113923.
Sequence in context: A006565 A026936 A021114 this_sequence A022738 A017269 A021079
Adjacent sequences: A113973 A113974 A113975 this_sequence A113977 A113978 A113979
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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 31 2006
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