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A132262
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First term in a sum partition of the even-indexed Fibonacci numbers.
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3
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1, 2, 7, 29, 130, 611, 2965, 14726, 74443, 381617, 1978582, 10355303, 54628201, 290148890, 1550177791, 8324934533, 44911554826, 243274479131, 1322555721037, 7213659006350, 39462884371891, 216470673634217, 1190382865461742, 6560913758341199
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OFFSET
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0,2
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COMMENTS
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This is the number in the center of the 3-regular tree when the exceptional representations of the 3-Kronecker quiver, whose dimension vector is given by subsequent even-indexed Fibonacci numbers are drawn into the 3-regular tree (the universal cover of the quiver).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ph. Fahr and C. M. Ringel, A Partition Formula for Fibonacci Numbers, preprint, 2007.
Ph. Fahr and C. M. Ringel, A Partition Formula for Fibonacci Numbers, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.4
Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, Journal of Integer Sequences, Vol. 15 (2012), Article 12.2.1
Michael D. Hirschhorn, On Recurrences of Fahr and Ringel Arising in Graph Theory, Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.8
Harris Kwong, On recurrences of Fahr and Ringel: an alternate approach, Fibonacci Quart. 48 (2010), no. 4, 363-365.
H. Prodinger, Generating functions related to partition formulas for Fibonacci Numbers, JIS 11 (2008) 08.1.8.
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FORMULA
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G.f.: (3*sqrt(1-6*q+q^2)-(1+q))/(2*(1-7*q+q^2)) = 1 +2q +7q^2 +29q^3 +130q^4 + ... . Michael David Hirschhorn, Sep 03 2009
a(n) ~ 3*sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n+1) / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 08 2014
Conjecture: n*a(n) + (-13*n+9)*a(n-1) + 22*(2*n-3)*a(n-2) + (-13*n+30)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Aug 28 2015
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EXAMPLE
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a(3) = 29 because 377 = 29 + 6*18 + 24*6 + 96*1.
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MAPLE
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a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 29][n+1],
((13*n-9)*a(n-1) -(44*n-66)*a(n-2)
+(13*n-30)*a(n-3) -(n-3)*a(n-4))/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 19 2013
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MATHEMATICA
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a[n_] := a[n] = If[n<4, {1, 2, 7, 29}[[n+1]], ((13*n-9)*a[n-1] - (44*n-66)*a[n-2] + (13*n-30)*a[n-3] - (n-3)*a[n-4])/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 07 2016, after Alois P. Heinz *)
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PROG
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(PARI) lista(nn) = my(q = qq + O(qq^nn)); gf = (3*sqrt(1-6*q+q^2) -(1+q))/(2*(1-7*q+q^2)); Vec(gf) \\ Michel Marcus, Jun 19 2013
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CROSSREFS
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Cf. A110122.
Sequence in context: A150664 A193040 A200755 * A007852 A300048 A232971
Adjacent sequences: A132259 A132260 A132261 * A132263 A132264 A132265
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KEYWORD
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nonn,easy
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AUTHOR
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Ph. Fahr and Claus Michael Ringel, Aug 19 2007
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EXTENSIONS
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More terms from Michel Marcus, Jun 19 2013
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STATUS
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approved
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