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

Author

Danny Rorabaugh, Mar 23 2015

Keywords

• nonn
• easy

Comments

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)

Examples

			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}.
		

Links

• 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).

Crossrefs

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).

Programs

Magma

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

Mathematica

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 *)

Sage

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

Formula

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)).