A213655 -id:A213655 - cp's OEIS frontend

A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.

0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330

Offset: 0

Danny Rorabaugh, Mar 23 2015

a(n) is also the number of ways to partition n into 2 or 3 parts.
a(n) is also the dimension of linear space of three-dimensional 2n-homogeneous polynomial vector fields, which have an octahedral symmetry (for a given representation), which are solenoidal, and which are vector fields on spheres. - Giedrius Alkauskas, Sep 30 2017
Apparently a(n) = A244239(n-6) for n > 4. - Georg Fischer, Oct 09 2018
a(n) is also the number of loopless connected n-regular multigraphs with 4 nodes. - Natan Arie Consigli, Aug 09 2019
a(n) is also the number of inequivalent linear [n, k=2] binary codes without 0 columns (see A034253 for more details). - Petros Hadjicostas, Oct 02 2019
Differs from A160138 only by the offset. - R. J. Mathar, May 15 2023
From Allan Bickle, Jul 13 2025: (Start)
a(n) is the number of theta graphs with n-2 vertices, or n-1 edges.  Equivalently, the number of 2-connected graphs with n-2 vertices and n-1 edges.
A theta graph has three paths with length at least 1 identified at their endpoints.  There can at most one path with length 1.
For instance the theta graphs with 6 vertices have paths with lengths (1,2,4), (1,3,3), or (2,2,2), so a(6-2) = 3. (End)

			On vertex set {a, b, c}, every connected multigraph with n = 5 edges is isomorphic to a multigraph with one of the following a(5) = 4 edge multisets: {ab, ab, ab, ab, ac}, {ab, ab, ab, ac, ac}, {ab, ab, ab, ac, bc}, and {ab, ab, ac, ac, bc}.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG], 2017; see Section 5.3.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k = 2.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,2,2}.]
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; see Eq. (23).
Gordon Royle, Small Multigraphs.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A003082, A014395, A014396, A014397, A014398, A050535, A076864.
Column k = 3 of A191646 and column k = 2 of A034253.
First differences of A034198 (excepting the first term).
Cf. A213654, A213655, A213668 (theta graphs).

[Floor(n/2) + Floor((n^2 + 6)/12): n in [0..70]]; // Vincenzo Librandi, Mar 24 2015

CoefficientList[Series[- x^2 (x^3 - x - 1) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Mar 24 2015 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 61] (* Robert G. Wilson v, Oct 11 2017 *)
a[n_]:=Floor[n/2] + Floor[(n^2 + 6)/12]; Array[a, 70, 0] (* Stefano Spezia, Oct 09 2018 *)

[floor(n/2) + floor((n^2 + 6)/12) for n in range(70)]

a(n) = A004526(n) + A069905(n).
a(n) = floor(n/2) + floor((n^2 + 6)/12).
G.f.: x^2*(x^3 - x - 1)/((x - 1)^2*(x^2 - 1)*(x^2 + x + 1)).

A213654 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the theta-graph TH(2,2,n) (n>=1, 1<=k<=n+3).

Original entry on oeis.org

2, 6, 4, 1, 0, 7, 10, 5, 1, 0, 3, 16, 15, 6, 1, 0, 2, 16, 30, 21, 7, 1, 0, 0, 12, 42, 50, 28, 8, 1, 0, 0, 5, 44, 87, 77, 36, 9, 1, 0, 0, 2, 33, 116, 158, 112, 45, 10, 1, 0, 0, 0, 19, 119, 253, 263, 156, 55, 11, 1, 0, 0, 0, 7, 96, 322, 488, 411, 210, 66, 12, 1

Offset: 1

Emeric Deutsch, Jun 18 2012

A theta-graph is a graph consisting of two vertices of degree three, connected by three paths of one or more edges each. In the theta-graph TH(2,2,n) the three paths have 2, 2, and n edges, respectively.
The entries in row n are the coefficients of the domination polynomial of the theta-graph TH(2,2,n) (see the Alikhani and Peng arxiv reference).
Sum of entries in row n is A213655(n).

			T(1,1)=2 because in the theta-graph TH(2,2,1) any of the two vertices of degree 3 is dominating.
Irregular triangle starts:
2,6,4,1;
0,7,10,5,1;
0,3,16,15,6,1;
0,2,16,30,21,7,1;

S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math., 30, No. 1, 2010, 37-51.

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.

Cf. A213655

p:=proc(n) if n = 1 then sort(x^4+4*x^3+6*x^2+2*x) elif n = 2 then sort(x^5+5*x^4+10*x^3+7*x^2) elif n = 3 then sort(x^6+6*x^5+15*x^4+16*x^3+3*x^2) else sort(expand(x*(p(n-1)+p(n-2)+p(n-3)))) end if end proc: for n to 13 do seq(coeff(p(n), x, k), k = 1 .. n+3) end do; # yields sequence in triangular form

p[n_] := p[n] = Switch[n, 1, x^4 + 4*x^3 + 6*x^2 + 2*x, 2, x^5 + 5*x^4 + 10*x^3 + 7*x^2, 3, x^6 + 6*x^5 + 15*x^4 + 16*x^3 + 3*x^2, _, Expand[x* (p[n - 1] + p[n - 2] + p[n - 3])]];
Table[CoefficientList[p[n], x] // Rest, {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 02 2017, from Maple *)

If p(n)=p(n,x) denotes the generating polynomial of row n (called the domination polynomial of the theta-graph TH(2,2,n), then p(n) = x*[p(n-1) + p(n-2) + p(n-3)] for n>=4; p(1), p(2), p(3) are given in the Maple program.

cp's OEIS Frontend

A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.

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