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A178134
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A generalized Lucas-binomial Fibonacci sequence based on A176263: a0=5 a(n)=Sum[A176263[n - m - 1, m, a0], {m, 0, Floor[(n - 1)/2]}]
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0
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0, 1, 1, 2, -3, -2, -32, -81, -311, -810, -2515, -6864, -19944, -55043, -156023, -433522, -1217427, -3391226, -9488456, -26462205, -73933535, -206293134, -576040339, -1607642688, -4488069168, -12526662167, -34967630447
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OFFSET
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0,4
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COMMENTS
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The limiting ratio is (alternating towards):2.7912878474779.
What makes this sequence interesting is that it is based on a generalization of
both Fibonacci types and Lucas -binomial Fibonacci types to give
what appears to be a new approach to ratio-ed sequences.
Both A000800,A004148 are this type of sequence, but where the Narayana (1,3,1)based one has ratio 1+Phi,
the Eulerian (1,4,1) based one never gets a steady ratio limit.
This sequence is an signed {1,4,1} type that gets a steady ratio limit.
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LINKS
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Table of n, a(n) for n=0..26.
Index to sequences with linear recurrences with constant coefficients, signature (1,7,-2,-6,-4,-25,5,25).
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FORMULA
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G.f. -x*(1-6*x^2-10*x^3-5*x^4+5*x^5) / ( (x-1)*(1+x)*(5*x^2+x-1)*(5*x^4+x^2-1) ). - R. J. Mathar, Nov 05 2012
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MATHEMATICA
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Clear[a, f, a0, t]
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
t[n_, m_, a_] := f[m + 1, a] + f[n + 1 - m, a] - f[n + 1, a];
a = 5;
a0[n_] := Sum[t[n - m - 1, m, a], {m, 0, Floor[(n - 1)/2]}];
Table[a0[n], {n, 0, 30}]
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CROSSREFS
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Cf. A000800, A004148
Sequence in context: A136454 A025522 A019228 * A075121 A075108 A190146
Adjacent sequences: A178131 A178132 A178133 * A178135 A178136 A178137
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KEYWORD
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uned,sign
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AUTHOR
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Roger L. Bagula, May 20 2010
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STATUS
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approved
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