A122047 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.

0, 0, 1, 3, 6, 10, 15, 22, 31, 42, 55, 70, 88, 109, 133, 160, 190, 224, 262, 304, 350, 400, 455, 515, 580, 650, 725, 806, 893, 986, 1085, 1190, 1302, 1421, 1547, 1680, 1820, 1968, 2124, 2288, 2460, 2640, 2829, 3027

Offset: 0

Author

Roger L. Bagula, Sep 13 2006

Keywords

• nonn

Comments

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

Links

• Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
• Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231 [quant-ph], 2008. [Eq 10a, lambda=5]

Crossrefs

Cf. A014125, A122046.

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.

Programs

Mathematica

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}]

Formula

Conjectures from R. J. Mathar, Jul 15 2008: (Start)

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);

o.g.f.: x^2/((x^4+x^3+x^2+x+1)(x-1)^4). (End)

Conjecture: a(n) = (A000292(n+1) - n - 2 - (-1)^floor((n-1)/5)*A099443(n+1))/5. - R. J. Mathar, Jul 15 2008

a(n+2) = A144679(n) + A144679(n-1) + A144679(n-2) + A144679(n-3) + A144679(n-4). - Johannes W. Meijer, May 20 2011

a(n) = floor((n^3 + 6*n^2 + 5*n)/30). - Allan Bickle, Sep 15 2022

Extensions

Edited by N. J. A. Sloane, Jul 15 2008

a(22)-a(43) from R. J. Mathar, Jul 15 2008