A000631 -id:A000631 - cp's OEIS frontend

A014125 Bisection of A001400.

1, 3, 6, 11, 18, 27, 39, 54, 72, 94, 120, 150, 185, 225, 270, 321, 378, 441, 511, 588, 672, 764, 864, 972, 1089, 1215, 1350, 1495, 1650, 1815, 1991, 2178, 2376, 2586, 2808, 3042, 3289, 3549, 3822, 4109, 4410, 4725, 5055, 5400, 5760, 6136, 6528, 6936, 7361, 7803

Offset: 0

N. J. A. Sloane

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

			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

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).

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 two-way infinite sequences
Index entries for covering numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A000217, A000631, A001840, A014126, A086161, A095708.
A column of A036838.
Cf. A002623, A014125, A122046, A122047. - Johannes W. Meijer, May 20 2011
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, A175724.

[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

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

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).

CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 14 2015 *)

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 */

my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ Altug Alkan, Oct 16 2015

a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ Charles R Greathouse IV, Jun 29 2020

[(binomial(n+4,3) - ((n+4)//3))/3 for n in (0..50)] # G. C. Greubel, Apr 28 2019

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)
a(n) = A144677(n) + A144677(n-1) + A144677(n-2).
a(n) = A190717(n-4) + 2*A190717(n-3) + 3*A190717(n-2) + 2*A190717(n-1) + A190717(n). (End)
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

More terms from James Sellers, Dec 24 1999

A027852 Number of connected functions on n points with a loop of length 2.

Original entry on oeis.org

0, 1, 1, 3, 6, 16, 37, 96, 239, 622, 1607, 4235, 11185, 29862, 80070, 216176, 586218, 1597578, 4370721, 12003882, 33077327, 91433267, 253454781, 704429853, 1962537755, 5479855546, 15332668869, 42983656210, 120716987723, 339596063606, 956840683968

Offset: 1

Christian G. Bower, Dec 14 1997

nonn

Number of unordered pairs of rooted trees with a total of n nodes.
Equivalently, the number of rooted trees on n+1 nodes where the root has degree 2.
Number of trees on n nodes rooted at an edge. - Washington Bomfim, Jul 06 2012
Guy (1988) calls these tadpole graphs. - N. J. A. Sloane, Nov 04 2014
Number of unicyclic graphs of n nodes with a cycle length of two (in other words, a double edge). - Washington Bomfim, Dec 02 2020

Alois P. Heinz, Table of n, a(n) for n = 1..2136
Washington Bomfim, Illustration of initial terms
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Eq. (75).
Index entries for sequences related to rooted trees

Column 2 of A033185 (forests of rooted trees), A217781 (unicyclic graphs), A339303 (unoriented linear forests) and A339428 (connected functions).

Cf. A000081, A000106, A000226, A000631, A001372, A002861.

with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> (add(b(i) *b(n-i), i=0..n) +`if`(irem(n, 2)=0, b(n/2), 0))/2: seq(a(n), n=1..50);  # Alois P. Heinz, Aug 22 2008, revised Oct 07 2011
# second, re-usable version
A027852 := proc(N::integer)
    local dh, Nprime;
    dh := 0 ;
    for Nprime from 0 to N do
        dh := dh+A000081(Nprime)*A000081(N-Nprime) ;
    end do:
    if type(N,'even') then
        dh := dh+A000081(N/2) ;
    end if;
    dh/2 ;
end proc: # R. J. Mathar, Mar 06 2017

Needs["Combinatorica`"];nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[2], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {2, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d b[d], {d, Divisors[j]}] b[n-j], {j, 1, n-1}])/(n-1)];
a[n_] := (Sum[b[i] b[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, b[n/2], 0])/2;
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 30 2018, after Alois P. Heinz *)

seq(max_n)= { my(V = f = vector(max_n), i=1,s); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n = 1, max_n, s = sum(k = 1, (n-1)/2, ( f[k] * f[n-k] ));
if(n % 2 == 1, V[i] = s, V[i] = s + (f[n/2]^2 + f[n/2])/2); i++); V };
\\ Washington Bomfim, Jul 06 2012 and Dec 01 2020

G.f.: A(x) = (B(x)^2 + B(x^2))/2 where B(x) is g.f. of A000081.
a(n) = Sum_{k=1..(n-1)/2}( f(k)*f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2+f(n/2) ) /2, where f(n) = A000081(n). - Washington Bomfim, Jul 06 2012 and Dec 01 2020
a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.43992401257102530404090339... . - Vaclav Kotesovec, Sep 12 2014
2*a(n) = A000106(n) + A000081(n/2), where A(.)=0 if the argument is non-integer. - R. J. Mathar, Jun 04 2020

Edited by Christian G. Bower, Feb 12 2002

A014126 Number of partitions of 2*n into at most 4 parts.

Original entry on oeis.org

1, 2, 5, 9, 15, 23, 34, 47, 64, 84, 108, 136, 169, 206, 249, 297, 351, 411, 478, 551, 632, 720, 816, 920, 1033, 1154, 1285, 1425, 1575, 1735, 1906, 2087, 2280, 2484, 2700, 2928, 3169, 3422, 3689, 3969, 4263, 4571, 4894, 5231, 5584, 5952, 6336, 6736, 7153, 7586

Offset: 0

N. J. A. Sloane

Bisection of A001400.
Molien series for 4-dimensional group of structure S_4 X C_2 and order 48, arising from complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4) containing the all-ones vector.
Partial sums of A156040. - Bob Selcoe, Feb 08 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
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)
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A000631, A001400, A014125, A049347, A092498, A156040.

with(combstruct): seq(count(Partition((2*n+4)), size=4), n=0..50); # Zerinvary Lajos, Mar 28 2008

CoefficientList[Series[(1 + x^2) / ((1 - x)^2 (1 - x^2) (1 - x^3)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,5,9,15,23,34},50] (* Harvey P. Dale, Aug 31 2015 *)

a(n)=(4*n^3+30*n^2+72*n+55+8*[1,-1,0][(n%3)+1]+9*(-1)^n)/72

G.f.: (1+x^2)/((1-x)^2*(1-x^2)*(1-x^3)). - James Sellers
a(n) = (1/72) * (4*n^3 + 30*n^2 + 72*n + 55 + 8*A049347(n) + 9*(-1)^n ). - Ralf Stephan, Aug 15 2013
E.g.f.: exp(-x)*(27 + 3*exp(2*x)*(55 + 106*x + 42*x^2 + 4*x^3) + 8*exp(x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/216. - Stefano Spezia, Apr 05 2023

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A014125 Bisection of A001400.

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A027852 Number of connected functions on n points with a loop of length 2.

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A014126 Number of partitions of 2*n into at most 4 parts.

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