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

%I A253186 #84 Jul 18 2025 21:45:56
%S A253186 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,
%T A253186 69,74,79,84,90,95,101,107,113,119,126,132,139,146,153,160,168,175,
%U A253186 183,191,199,207,216,224,233,242,251,260,270,279,289,299,309,319,330
%N A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.
%C A253186 a(n) is also the number of ways to partition n into 2 or 3 parts.
%C A253186 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
%C A253186 Apparently a(n) = A244239(n-6) for n > 4. - _Georg Fischer_, Oct 09 2018
%C A253186 a(n) is also the number of loopless connected n-regular multigraphs with 4 nodes. - _Natan Arie Consigli_, Aug 09 2019
%C A253186 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
%C A253186 Differs from A160138 only by the offset. - _R. J. Mathar_, May 15 2023
%C A253186 From _Allan Bickle_, Jul 13 2025: (Start)
%C A253186 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.
%C A253186 A theta graph has three paths with length at least 1 identified at their endpoints.  There can at most one path with length 1.
%C A253186 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)
%H A253186 Danny Rorabaugh, <a href="/A253186/b253186.txt">Table of n, a(n) for n = 0..10000</a>
%H A253186 Giedrius Alkauskas, <a href="https://arxiv.org/abs/1601.06570">Projective and polynomial superflows. I</a>, arxiv.org/1601.06570 [math.AG], 2017; see Section 5.3.
%H A253186 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A253186 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_4.html">Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns</a>. [See column k = 2.]
%H A253186 H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. 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}.]
%H A253186 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000  [math.CO], 2017; see Eq. (23).
%H A253186 Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/graphs/multigraphs/index.html">Small Multigraphs</a>.
%H A253186 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A253186 a(n) = A004526(n) + A069905(n).
%F A253186 a(n) = floor(n/2) + floor((n^2 + 6)/12).
%F A253186 G.f.: x^2*(x^3 - x - 1)/((x - 1)^2*(x^2 - 1)*(x^2 + x + 1)).
%e A253186 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}.
%t A253186 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 *)
%t A253186 LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 61] (* _Robert G. Wilson v_, Oct 11 2017 *)
%t A253186 a[n_]:=Floor[n/2] + Floor[(n^2 + 6)/12]; Array[a, 70, 0] (* _Stefano Spezia_, Oct 09 2018 *)
%o A253186 (Sage) [floor(n/2) + floor((n^2 + 6)/12) for n in range(70)]
%o A253186 (Magma) [Floor(n/2) + Floor((n^2 + 6)/12): n in [0..70]]; // _Vincenzo Librandi_, Mar 24 2015
%Y A253186 Cf. A001399, A003082, A014395, A014396, A014397, A014398, A050535, A076864.
%Y A253186 Column k = 3 of A191646 and column k = 2 of A034253.
%Y A253186 First differences of A034198 (excepting the first term).
%Y A253186 Cf. A213654, A213655, A213668 (theta graphs).
%K A253186 nonn,easy
%O A253186 0,4
%A A253186 _Danny Rorabaugh_, Mar 23 2015