|
%I N1005 #144 Feb 26 2024 01:30:24
%S 1,3,6,11,18,27,39,54,72,94,120,150,185,225,270,321,378,441,511,588,
%T 672,764,864,972,1089,1215,1350,1495,1650,1815,1991,2178,2376,2586,
%U 2808,3042,3289,3549,3822,4109,4410,4725,5055,5400,5760,6136,6528,6936,7361,7803
%N Bisection of A001400.
%C Also Schoenheim bound L_1(n,5,4).
%C Degrees of polynomials defined by p(n) = (x^(n+1)*p(n-1)p(n-3) + p(n-2)^2)/p(n-4), p(-4)=p(-3)=p(-2)=p(-1)=1. - _Michael Somos_, Jul 21 2004
%C Degrees of polynomial tau-functions of q-discrete Painlevé I, which generate sequence A095708 when q=2 (up to an offset of 3). - _Andrew Hone_, Jul 29 2004
%C Because of the Laurent phenomenon for the general q-discrete Painlevé I tau-function recurrence p(n) = (a*x^(n+1)*p(n-1)*p(n-3) + b*p(n-2)^2)/p(n-4), p(n) for n > -1 will always be a polynomial in x and a Laurent polynomial in a,b and the initial data p(-4),p(-3),p(-2),p(-1). - _Andrew Hone_, Jul 29 2004
%C Create the sequence 0,0,0,0,0,6,18,36,66,108,... so that the sum of three consecutive terms b(n) + b(n+1) + b(n+2) = A007531(n), with b(0)=0; then a(n) = b(n+5)/6. - _J. M. Bergot_, Jul 30 2013
%C Number of partitions of n into three kinds of part 1 and one kind of part 3. - _Joerg Arndt_, Sep 28 2015
%C First differences are A001840(k) starting with k=2; second differences are A086161(k) starting with k=1. - _Bob Selcoe_, Sep 28 2015
%C Maximum Wiener index of all maximal planar graphs with n+2 vertices. The extremal graphs are cubes of paths. - _Allan Bickle_, Jul 09 2022
%C Maximum Wiener index of all maximal 3-degenerate graphs with n+2 vertices. (A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices.) The extremal graphs are cubes of paths, so the bound also applies to 3-trees. - _Allan Bickle_, Sep 18 2022
%D W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992. See Eq. 1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D L. Smiley, Hidden Hexagons (preprint).
%H Michael De Vlieger, <a href="/A014125/b014125.txt">Table of n, a(n) for n = 0..10000</a>
%H Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, <a href="https://doi.org/10.1007/s10801-015-0647-5">Laurent phenomenon sequences</a>, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
%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 Z. Che and K. L. Collins, <a href="https://doi.org/10.1016/j.dam.2018.11.026">An upper bound on Wiener indices of maximal planar graphs</a>, Discrete Appl. Math. 258 (2019), 76-86.
%H Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, <a href="https://arxiv.org/abs/1905.06753">Wiener Index and Remoteness in Triangulations and Quadrangulations</a>, arXiv:1905.06753 [math.CO], 2019.
%H S. Fomin and A. Zelevinsky, <a href="http://dx.doi.org/10.1006/aama.2001.0770">The Laurent Phenomenon</a>, Advances in Applied Mathematics, 28 (2002), 119-144.
%H D. Ghosh, E. Győri, A. Paulos, N. Salia, and O. Zamora, <a href="https://doi.org/10.1007/s10878-020-00655-4">The maximum Wiener index of maximal planar graphs</a>, Journal of Combinatorial Optimization 40, (2020), 1121-1135.
%H H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01329a033">The number of structurally isomeric hydrocarbons of the ethylene series</a>, J. Amer. Chem. Soc., 55 (1933), 680-685.
%H H. R. Henze and C. M. Blair, <a href="/A000631/a000631.pdf">The number of structurally isomeric Hydrocarbons of the Ethylene Series</a>, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
%H A. N. W. Hone, <a href="http://arXiv.org/abs/0807.2538">Algebraic curves, integer sequences and a discrete Painlevé transcendent</a>, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [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 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=3]
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Cor#covnum">Index entries for covering numbers</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F G.f.: 1/((1-x)^3*(1-x^3)).
%F a(n) = -a(-6-n) = 3*a(n-1) -3*a(n-2) +2*a(n-3) -3*a(n-4) +3*a(n-5) -a(n-6).
%F The simplest recurrence is fourth order: a(n) = a(n-1) + a(n-3) - a(n-4) + n + 1, which gives the g.f.: 1/((1-x)^3*(1-x^3)), with a(-4) = a(-3) = a(-2) = a(-1) = 0.
%F a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + 2/(9*sqrt(3))*sin(2*Pi*n/3). - _Andrew Hone_, Jul 29 2004
%F a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + (((n+1) mod 3) - 1)/9. - same formula, simplified by _Gerald Hillier_, Apr 14 2015
%F a(n) = (2*A000027(n+1) + 3*A000292(n+1) + A049347(n-1) + 1 + 3*A000217(n+1))/9. - _R. J. Mathar_, Nov 16 2007
%F From _Johannes W. Meijer_, May 20 2011: (Start)
%F a(n) = A144677(n) + A144677(n-1) + A144677(n-2).
%F a(n) = A190717(n-4) + 2*A190717(n-3) + 3*A190717(n-2) + 2*A190717(n-1) + A190717(n). (End)
%F 3*a(n) = binomial(n+4,3) - floor((n+4)/3). - _Bruno Berselli_, Nov 08 2013
%F a(n) = A000217(n+1) + a(n-3) = Sum_{j>=0, n>=3*j} (n-3*j+1)*(n-3*j+2)/2. - _Bob Selcoe_, Sep 27 2015
%F a(n) = round(((2*n+5)^3 + 3*(2*n+5)^2 - 9*(2*n+5))/144). - _Giacomo Guglieri_, Jun 28 2020
%F a(n) = floor(((n+2)^3 + 3*(n+2)^2)/18). - _Allan Bickle_, Aug 01 2020
%F a(n) = Sum_{j=0..n} (n-j+1)*floor((j+3)/3). - _G. C. Greubel_, Oct 18 2021
%F E.g.f.: exp(x) + exp(x)*x*(34 + 12*x + x^2)/18 + 2*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - _Stefano Spezia_, Apr 05 2023
%e Polynomials: p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...
%e a(12)=185: A000217(13)=91 + a(9)=94 == 91+55+28+10+1 = 185. - _Bob Selcoe_, Sep 27 2015
%e a(3)=11: the 11 partitions of 3 are {1a,1a,1a}, {1a,1a,1b}, {1a,1a,1c}, {1a,1b,1b}, {1a,1b,1c}, {1a,1c,1c}, {1b,1b,1b}, {1b,1b,1c}, {1b,1c,1c}, {1c,1c,1c}, {3}. - _Bob Selcoe_, Oct 04 2015
%p L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Current sequence is L_1(n,n-3,n-4,1).
%t CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Apr 14 2015 *)
%o (PARI) a(n)=if(n<-5,-a(-6-n),polcoeff(1/(1-x)^3/(1-x^3)+x^n*O(x),n)) /* _Michael Somos_, Jul 21 2004 */
%o (PARI) my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ _Altug Alkan_, Oct 16 2015
%o (PARI) a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ _Charles R Greathouse IV_, Jun 29 2020
%o (Magma) [n^3/18+n^2/2+4*n/3+1+(((n+1) mod 3)-1)/9 : n in [0..50]]; // _Wesley Ivan Hurt_, Apr 14 2015
%o (Magma) I:=[1,3,6,11,18,27]; [n le 6 select I[n] else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3)-3*Self(n-4)+3*Self(n-5)-Self(n-6): n in [1..50]]; // _Vincenzo Librandi_, Apr 15 2015
%o (Sage) [(binomial(n+4,3) - ((n+4)//3))/3 for n in (0..50)] # _G. C. Greubel_, Apr 28 2019
%Y Cf. A000217, A000631, A001840, A014126, A086161, A095708.
%Y A column of A036838.
%Y Cf. A002623, A014125, A122046, A122047. - _Johannes W. Meijer_, May 20 2011
%Y Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, A175724.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 24 1999
|
