OFFSET
0,2
COMMENTS
Also Schoenheim bound L_1(n,5,4).
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
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
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
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
Number of partitions of n into three kinds of part 1 and one kind of part 3. - Joerg Arndt, Sep 28 2015
First differences are A001840(k) starting with k=2; second differences are A086161(k) starting with k=1. - Bob Selcoe, Sep 28 2015
Maximum Wiener index of all maximal planar graphs with n+2 vertices. The extremal graphs are cubes of paths. - Allan Bickle, Jul 09 2022
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
REFERENCES
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
L. Smiley, Hidden Hexagons (preprint).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10000
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
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.
Z. Che and K. L. Collins, An upper bound on Wiener indices of maximal planar graphs, Discrete Appl. Math. 258 (2019), 76-86.
Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, Wiener Index and Remoteness in Triangulations and Quadrangulations, arXiv:1905.06753 [math.CO], 2019.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
D. Ghosh, E. Győri, A. Paulos, N. Salia, and O. Zamora, The maximum Wiener index of maximal planar graphs, Journal of Combinatorial Optimization 40, (2020), 1121-1135.
H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [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 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=3]
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).
FORMULA
G.f.: 1/((1-x)^3*(1-x^3)).
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).
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.
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
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
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
From Johannes W. Meijer, May 20 2011: (Start)
3*a(n) = binomial(n+4,3) - floor((n+4)/3). - Bruno Berselli, Nov 08 2013
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
a(n) = round(((2*n+5)^3 + 3*(2*n+5)^2 - 9*(2*n+5))/144). - Giacomo Guglieri, Jun 28 2020
a(n) = floor(((n+2)^3 + 3*(n+2)^2)/18). - Allan Bickle, Aug 01 2020
a(n) = Sum_{j=0..n} (n-j+1)*floor((j+3)/3). - G. C. Greubel, Oct 18 2021
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
EXAMPLE
Polynomials: p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...
a(12)=185: A000217(13)=91 + a(9)=94 == 91+55+28+10+1 = 185. - Bob Selcoe, Sep 27 2015
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
MAPLE
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).
MATHEMATICA
CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 14 2015 *)
PROG
(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 */
(PARI) my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ Altug Alkan, Oct 16 2015
(PARI) a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ Charles R Greathouse IV, Jun 29 2020
(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
(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
(Sage) [(binomial(n+4, 3) - ((n+4)//3))/3 for n in (0..50)] # G. C. Greubel, Apr 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Dec 24 1999
STATUS
approved