A126201 -id:A126201 - cp's OEIS frontend

A005470 Number of unlabeled planar simple graphs with n nodes.

1, 1, 2, 4, 11, 33, 142, 822, 6966, 79853, 1140916, 18681008, 333312451

Offset: 0

Euler transform of A003094. - Christian G. Bower

			a(2) = 2 since o o and o-o are the two planar simple graphs on two nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Trotter, ed., Planar Graphs, Vol. 9, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., 1993.
Turner, James; Kautz, William H. A survey of progress in graph theory in the Soviet Union. SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 19. - N. J. A. Sloane, Apr 08 2014
Vetukhnovskii, F. Ya. "Estimate of the Number of Planar Graphs." In Soviet Physics Doklady, vol. 7, pp. 7-9. 1962. - From N. J. A. Sloane, Apr 08 2014
R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

G. Brinkmann, and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007) 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
E. Friedman, Illustration of small graphs
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Planar Graph
Index entries for "core" sequences

Cf. A003094 (connected planar graphs), A034889, A039735 (planar graphs by nodes and edges).

Cf. A126201.

A003094 = Cases[Import["https://oeis.org/A003094/b003094.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A003094] (* Jean-François Alcover, Apr 25 2013, updated Mar 17 2020 *)

n=8 term corrected and n=9..11 terms calculated by Brendan McKay
Terms a(0) - a(10) confirmed by David Applegate and N. J. A. Sloane, Mar 09 2007
a(12) added by Vaclav Kotesovec after A003094 (computed by Brendan McKay), Dec 06 2014

A003094 Number of unlabeled connected planar simple graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 6, 20, 99, 646, 5974, 71885, 1052805, 17449299, 313372298, 5942258308

Offset: 0

N. J. A. Sloane

Inverse Euler transform of A005470. - Christian G. Bower, May 16 2003

			a(3) = 2 since the path o-o-o and the triangle are the two connected planar simple graphs on three nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. J. Wilson, Introduction to Graph Theory, Academic Press, NY, 1972, p. 162.

David Wasserman, Brendan McKay and Georg Grasegger, Table of n, a(n) for n = 0..13
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
E. Friedman, Illustration of small graphs
Brendan McKay, Planar graphs
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Planar Connected Graph

Row sums of A049334.
The labeled version is A096332.
Cf. A005470, A126201.

a[n_Integer?NonNegative] := a[n] = Module[{m, s, g}, s = Subsets[Range[n], {2}]; m = Length[s]; g = Graph[Range[n], UndirectedEdge @@@ #] & /@ (Pick[s, #, 1] & /@ (IntegerDigits[#, 2, m] & /@ Range[0, 2^m - 1])); Length[DeleteDuplicates[Select[Select[g, ConnectedGraphQ], PlanarGraphQ], IsomorphicGraphQ]]]; Table[a[n], {n, 0, 6}] (* Robert P. P. McKone, Oct 14 2023 *)

geng -c $n | planarg -q | countg -q # Georg Grasegger, Jul 06 2023

More terms from Brendan McKay
a(12) added by Brendan McKay, Dec 06 2014
a(13) added by Georg Grasegger, Jul 06 2023

A126100 Number of rooted connected unlabeled graphs on n nodes.

Original entry on oeis.org

0, 1, 1, 3, 11, 58, 407, 4306, 72489, 2111013, 111172234, 10798144310, 1944301471861, 650202565436890, 404697467417019634, 470133531223369393920, 1022561022228933341815171, 4177761667636803276899047351, 32163582481439081597751699343141, 468019937132164016636736323752098741

Offset: 0

David Applegate and N. J. A. Sloane, Mar 05 2007

Let G run through all connected unlabeled graphs on n nodes. Add up the numbers of inequivalent nodes (under Aut(G)) for each G.
"Pointed" connected graphs. This has the same relation to A001349 as A000081 does to A000055.
a(0) = 0 because the empty graph cannot be rooted.

			For 3 nodes G is either a path (2 kinds of nodes) or a triangle (one kind of node), for a total of a(3) = 3.
For the 5-vertex graphs we have 2 * 1 orbit, 6 * 2 orbits, 8 * 3 orbits, 5 * 4 orbits for a total of 2 + 12 + 24 + 20 = 58.

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from David Applegate and N. J. A. Sloane)

Cf. A001349, A126101, A000666, A000088, A126201, A303831 (birooted), A304311.

Cf. A000081, A000055.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
g[n_, r_] := (s = 0; Do[s += permcount[p]*(2^(r*Length[p] + edges[p])), {p, IntegerPartitions[n]}]; s/n!);
seq[m_] := Sum[g[n-1, 1] x^(n-1), {n, 0, m}]/Sum[g[n-1, 0] x^(n-1), {n, 0, m}] + O[x]^m // CoefficientList[#, x]& // Prepend[#, 0]&;
seq[20] (* Jean-François Alcover, Jul 09 2018, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2)}
g(n,r) = {my(s=0); forpart(p=n, s+=permcount(p)*(2^(r*#p+edges(p)))); s/n!}
seq(n)={concat([0], Vec(Ser(vector(n, n, g(n-1,1)))/Ser(vector(n, n, g(n-1,0)))))} \\ Andrew Howroyd, May 03 2018

The g.f. A(x) = x+x^2+3*x^3+11*x^4+... satisfies f(x) = 1 + A(x)*g(x), where f(x) = 1+x+2*x^2+6*x^3+20*x^4+... is the g.f. for A000666 and g(x) = 1+x+2*x^2+4*x^3+11*x^4+... is the g.f. for A000088. - Brendan McKay

a(5)-a(9) computed by Gordon F. Royle, Mar 05 2007
a(10) and a(11) computed by Brendan McKay, Mar 05 2007
a(12) onwards computed from the generating function, A000088 and A000666 by David Applegate and N. J. A. Sloane, Mar 06 2007

A173422 Partials sums of A003094 (unlabeled connected planar simple graphs with n nodes).

Original entry on oeis.org

1, 2, 3, 5, 11, 31, 130, 776, 6750, 78635, 1131440, 18580739, 331953037

Offset: 0

Jonathan Vos Post, Feb 18 2010

Partials sums of number of unlabeled connected planar simple graphs with n nodes. The subsequence of primes in these partial sums begins: 2, 3, 5, 11, 31.

			a(11) = 1 + 1 + 1 + 2 + 6 + 20 + 99 + 646 + 5974 + 71885 + 1052805 + 17449299.

Cf. A003094, inverse Euler transform of A005470, A126201.

S=0; A173422=apply(t->S+=t,A003094) \\ with A003094 defined as vector of the initial terms of that sequence. - M. F. Hasler, Mar 20 2018

a(n) = Sum_{i=0..n} A003094(i).

Edited and a(12) added by M. F. Hasler, Mar 20 2018

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A005470 Number of unlabeled planar simple graphs with n nodes.

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A003094 Number of unlabeled connected planar simple graphs with n nodes.

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A126100 Number of rooted connected unlabeled graphs on n nodes.

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A173422 Partials sums of A003094 (unlabeled connected planar simple graphs with n nodes).

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