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A147953 A Fernandez-type expansion for modulo three recursion: f(n) = f(f(n - 2)) + 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)=Product[x+f(n),{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)). 0
1, 1, 1, 3, 4, 7, 9, 14, 22, 32, 43, 61, 89, 118, 167, 235, 312, 417, 572, 748, 1006, 1326, 1744, 2283, 2982, 3878, 5048, 6518, 8355, 10786, 13727, 17436, 22173, 28250, 35561, 45008, 56651, 70818, 88992, 111280, 138431, 172284, 214019, 265166, 328127 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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:1.1662119633190497 or smaller: Table[N[a2[[n + 1]]/a2[[n]]], {n, 1, 99}]. That ratio is lower than for Lehmer's degree 10 Salem.

LINKS

Table of n, a(n) for n=0..44.

FORMULA

f(n) = f(f(n - 2)) + 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)=Product[x+f(n),{n,0,Infinity}]; a(n)=Coefficients[(p(x,n)).

MATHEMATICA

f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 2]] + 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}]

CROSSREFS

A147665

Sequence in context: A266769 A008763 A005896 * A163468 A069183 A119907

Adjacent sequences:  A147950 A147951 A147952 * A147954 A147955 A147956

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Nov 17 2008

STATUS

approved

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)