OFFSET
0,2
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).
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.
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
EXAMPLE
a(3) = 29 because 377 = 29 + 6*18 + 24*6 + 96*1.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ph. Fahr and Claus Michael Ringel, Aug 19 2007
STATUS
approved