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%I #41 Feb 26 2024 01:30:50
%S 0,0,1,3,6,10,15,22,31,42,55,70,88,109,133,160,190,224,262,304,350,
%T 400,455,515,580,650,725,806,893,986,1085,1190,1302,1421,1547,1680,
%U 1820,1968,2124,2288,2460,2640,2829,3027
%N Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)*P(n-1,x)*P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.
%C Maximum Wiener index of all maximal 5-degenerate graphs with n vertices. (A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to 5 existing vertices.) The extremal graphs are 5th powers of paths, so the bound also applies to 5-trees. - _Allan Bickle_, Sep 15 2022
%H Allan Bickle and Zhongyuan Che, <a href="https://arxiv.org/abs/1908.09202">Wiener indices of maximal k-degenerate graphs</a>, arXiv:1908.09202 [math.CO], 2019.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H A. N. W. Hone, <a href="http://arXiv.org/abs/0807.2538">Algebraic curves, integer sequences and a discrete Painlevé transcendent</a>, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231 [quant-ph], 2008. [Eq 10a, lambda=5]
%F Conjectures from _R. J. Mathar_, Jul 15 2008: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8);
%F o.g.f.: x^2/((x^4+x^3+x^2+x+1)(x-1)^4). (End)
%F Conjecture: a(n) = (A000292(n+1) - n - 2 - (-1)^floor((n-1)/5)*A099443(n+1))/5. - _R. J. Mathar_, Jul 15 2008
%F a(n+2) = A144679(n) + A144679(n-1) + A144679(n-2) + A144679(n-3) + A144679(n-4). - _Johannes W. Meijer_, May 20 2011
%F a(n) = floor((n^3 + 6*n^2 + 5*n)/30). - _Allan Bickle_, Sep 15 2022
%t p[n_] := p[n] = Cancel[Simplify[(x^(n - 1)p[n - 1]p[n - 5] + p[n - 2]*p[n - 4])/p[n - 6]]];p[ -6] = 1; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
%Y Cf. A014125, A122046.
%Y The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623, A014125, A122046, A122047 (this sequence), A175724, respectively.
%K nonn
%O 0,4
%A _Roger L. Bagula_, Sep 13 2006
%E Edited by _N. J. A. Sloane_, Jul 15 2008
%E a(22)-a(43) from _R. J. Mathar_, Jul 15 2008
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