A132262 First term in a sum partition of the even-indexed Fibonacci numbers.

1, 2, 7, 29, 130, 611, 2965, 14726, 74443, 381617, 1978582, 10355303, 54628201, 290148890, 1550177791, 8324934533, 44911554826, 243274479131, 1322555721037, 7213659006350, 39462884371891, 216470673634217, 1190382865461742, 6560913758341199

Offset: 0

Author

Ph. Fahr and Claus Michael Ringel, Aug 19 2007

Keywords

• nonn
• easy

Comments

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).

Examples

			a(3) = 29 because 377 = 29 + 6*18 + 24*6 + 96*1.
		

Links

• 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.
• Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras, arXiv:2104.00050 [math.RT], 2021.
• 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.
• A. Moreno Canadas, P. F. Fernandez Espinoza, I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
• H. Prodinger, Generating functions related to partition formulas for Fibonacci Numbers, JIS 11 (2008) 08.1.8.

Crossrefs

Cf. A110122.

Programs

Maple

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

Mathematica

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 *)

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

Formula

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

D-finite with recurrence 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