A343869 -id:A343869 - cp's OEIS frontend

A002840 Number of polyhedral graphs with n edges.

1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, 485704, 1645576, 5623571, 19358410, 67078828, 233800162, 819267086, 2884908430, 10204782956, 36249143676, 129267865144, 462669746182, 1661652306539, 5986979643542

Offset: 6

N. J. A. Sloane

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Math. Comp. 37 (1981), no. 156, 523-532.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
G. P. Michon, Counting Polyhedra - Numericana
Hugo Pfoertner, Unlabeled 3-connected planar graphs for n<=20 edges, list in PARI-readable format.
Eric Weisstein's World of Mathematics, Polyhedral Graph
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Column sums of A049337.

Cf. A002841, A000944, A046091, A338511, A343869, A343871.

\\ It is assumed that the 3cp.gp file (from the linked zip archive) has been read before, i.e., \r [path]3cp.gp
for(k=6,#ThreeConnectedData,print1(#ThreeConnectedData[k],", "));
\\ printing of the edge lists of the graphs for n <= 11
print(ThreeConnectedData[6..11]) \\ Hugo Pfoertner, Feb 14 2021

a(30)-a(35) from the Numericana link added by Andrey Zabolotskiy, Jun 13 2020

A046091 Number of connected planar graphs with n edges.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 30, 79, 227, 709, 2318, 8049, 29372, 112000, 444855, 1833072, 7806724, 34252145, 154342391, 712231465, 3357126655, 16119421175, 78580665333

Offset: 0

Brendan McKay

Inverse Euler transform of A343872. - Andrew Howroyd, May 05 2021

			a(3) = 3 since the three connected graphs with three edges are a path, a triangle and a "Y".
The first difference between this sequence and A002905 is for n=9 edges where we see K_{3,3}, the "utility graph".

B. D. McKay and A. Piperno, Practical Graph Isomorphism, II, J. Symbolic Computation, 60 (2014), pp. 94-112.
Eric Weisstein's World of Mathematics, Planar Connected Graph.

Row sums of A343873.
Column sums of A049334.
Cf. A002905, A003094, A066951, A291842, A343869, A343872.

# count graphs for the sequence by number of vertices v, sum over v afterwards
geng -c $v $n:$n | planarg -q | countg -q # Georg Grasegger, Jul 06 2023

a(11)-a(19) from Martin Fuller using nauty by Brendan McKay, Mar 07 2015

a(20)-a(22) added by Georg Grasegger, Jul 06 2023

A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 3, 3, 1, 0, 0, 0, 0, 2, 9, 4, 1, 0, 0, 0, 0, 1, 13, 20, 6, 1, 0, 0, 0, 0, 0, 11, 49, 40, 7, 1, 0, 0, 0, 0, 0, 5, 77, 158, 70, 9, 1, 0, 0, 0, 0, 0, 2, 75, 406, 426, 121, 11, 1, 0, 0, 0, 0, 0, 0, 47, 662, 1645, 1018, 189, 13, 1, 0

Offset: 1

Andrew Howroyd, May 04 2021

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
  1;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 0, 1, 1,  0;
  0, 0, 1, 2,  1,  0;
  0, 0, 0, 3,  3,  1,   0;
  0, 0, 0, 2,  9,  4,   1,   0;
  0, 0, 0, 1, 13, 20,   6,   1,   0;
  0, 0, 0, 0, 11, 49,  40,   7,   1,  0;
  0, 0, 0, 0,  5, 77, 158,  70,   9,  1, 0;
  0, 0, 0, 0,  2, 75, 406, 426, 121, 11, 1, 0;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..351 (26 rows) (first 210 terms (20 rows) from Andrew Howroyd)

Row sums are A343869.
Column sums are A021103.
Cf. A049334, A049336 (transpose), A049337, A253186, A339070.

geng -C $k $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

T(n, n) = 1 for n >= 3.

T(n, n-1) = A253186(n-3) for n >= 3.

A289471 Number of planar strictly 2-connected graphs on n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 15, 39, 106, 316, 1030, 3553, 13006, 49780, 197281, 804649, 3358885, 14289507, 61769577

Offset: 1

Ed Pegg Jr, Jul 06 2017

Cf. A002218, A002840, A289470, A343869.

a(n) = A343869(n) - A002840(n). - Andrew Howroyd, May 04 2021

a(12)-a(13) corrected and a(14)-a(20) from Andrew Howroyd, May 04 2021

cp's OEIS Frontend

A002840 Number of polyhedral graphs with n edges.

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A046091 Number of connected planar graphs with n edges.

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A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

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A289471 Number of planar strictly 2-connected graphs on n edges.

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