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%I #49 Mar 25 2022 10:48:02
%S 1,2,7,29,130,611,2965,14726,74443,381617,1978582,10355303,54628201,
%T 290148890,1550177791,8324934533,44911554826,243274479131,
%U 1322555721037,7213659006350,39462884371891,216470673634217,1190382865461742,6560913758341199
%N First term in a sum partition of the even-indexed Fibonacci numbers.
%C 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).
%H Alois P. Heinz, <a href="/A132262/b132262.txt">Table of n, a(n) for n = 0..1000</a>
%H Ph. Fahr and C. M. Ringel, <a href="http://www.math.uni-bielefeld.de/~ringel/publ-new.html">A Partition Formula for Fibonacci Numbers</a>, preprint, 2007.
%H Ph. Fahr and C. M. Ringel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Fahr/ringel44.html">A Partition Formula for Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.4
%H Philipp Fahr and Claus Michael Ringel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Ringel/ringel10.html">Categorification of the Fibonacci Numbers Using Representations of Quivers</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.2.1.
%H Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, <a href="https://arxiv.org/abs/2104.00050">Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras</a>, arXiv:2104.00050 [math.RT], 2021.
%H Michael D. Hirschhorn, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Hirschhorn/hirschhorn8.html">On Recurrences of Fahr and Ringel Arising in Graph Theory</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.8
%H Harris Kwong, <a href="http://www.fq.math.ca/Papers1/48-4/Kwong.pdf">On recurrences of Fahr and Ringel: an alternate approach</a>, Fibonacci Quart. 48 (2010), no. 4, 363-365.
%H A. Moreno Canadas, P. F. Fernandez Espinoza, I. D. M. Gaviria, <a href="http://dx.doi.org/10.17654/NT038040339"">Categorification of some integer sequences via Kronecker Modules</a>, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
%H H. Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Prodinger/prodinger24.html">Generating functions related to partition formulas for Fibonacci Numbers</a>, JIS 11 (2008) 08.1.8.
%F 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
%F 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
%F 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
%e a(3) = 29 because 377 = 29 + 6*18 + 24*6 + 96*1.
%p a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 29][n+1],
%p ((13*n-9)*a(n-1) -(44*n-66)*a(n-2)
%p +(13*n-30)*a(n-3) -(n-3)*a(n-4))/n)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 19 2013
%t 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_ *)
%o (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
%Y Cf. A110122.
%K nonn,easy
%O 0,2
%A Ph. Fahr and _Claus Michael Ringel_, Aug 19 2007
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