A338511 -id:A338511 - cp's OEIS frontend

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4039, 12044, 36406, 111324, 342447, 1064835, 3341434, 10583931, 33728050, 107931849, 346616201

Offset: 0

Vaclav Kotesovec, Sep 02 2010

In "addendum" J. Karnofsky stated the value a(15) = 1064833. In contrast to the terms up to and including a(14), which could all be confirmed, an independent calculation based on a list of 3-connected simple graphs resulted in the corrected value a(15) = 1064835. - Hugo Pfoertner, Dec 06 2020

See A337517 for the number of different resistances that can be obtained by combining /exactly/ n one-ohm resistors. The method used by Andrew Howroyd (see his program in the link section) uses 3-connected graphs with one edge (the 'battery edge') removed. - Rainer Rosenthal, Feb 07 2021

			a(n) counts all resistances that can be obtained with fewer than n resistors as well as with exactly n resistors. Without a resistor the resistance is infinite, i.e., a(0) = 1. One 1-ohm resistor adds resistance 1, so a(1) = 2. Two resistors in parallel give 1/2 ohm, while in series they give 2 ohms. So a(2) is the number of elements in the set {infinity, 1, 1/2, 2}, i.e., a(2) = 4. - _Rainer Rosenthal_, Feb 07 2021

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Andrew Howroyd, PARI program (includes scripts for other related sequences)
Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Joel Karnofsky, Mathematica program for resistances, copied from article.
Index to sequences related to resistances.

Cf. A048211, A051389, A174284, A337517, A338197, A338487, A338573, A338580, A338590, A338600.

Cf. A123545, A338511, A338999, A339072.

Mathematica
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(* See link. *)
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a(n) = A174284(n) + 1 for n <= 7, a(n) > A174284(n) + 1 otherwise. - Hugo Pfoertner, Nov 01 2020

a(n) is the number of elements in the union of the sets SetA337517(k), k <= n, counted by A337517. - Rainer Rosenthal, Feb 07 2021

a(15) corrected and a(16) added by Hugo Pfoertner, Dec 06 2020
a(17) from Hugo Pfoertner, Dec 09 2020
a(0) from Rainer Rosenthal, Feb 07 2021
a(18) from Hugo Pfoertner, Apr 09 2021
a(19) from Zhao Hui Du, May 15 2023
a(20) from Zhao Hui Du, May 23 2023

A002840 Number of polyhedral graphs with n edges.

Original entry on oeis.org

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

A006290 Number of 3-connected graphs with n nodes.

Original entry on oeis.org

1, 3, 17, 136, 2388, 80890, 5114079, 573273505, 113095167034, 39582550575765, 24908445793058442, 28560405143495819079, 60364410130177223014724, 237403933018799958309530349, 1750323137355778190158082029500, 24333358813699371350715221107464003, 640811613278752754485012443963579501421

Offset: 4

N. J. A. Sloane

Robinson and Walsh list first 25 terms.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 4..25
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
T. R. S. Walsh, Counting unlabeled three-connected and homeomorphically irreducible two-connected graphs, J. Combin. Theory Ser. B 32 (1982), no. 1, 12-32.
Eric Weisstein's World of Mathematics, k-Connected Graph
David Kofoed Wind, Connected Graphs with Fewest Spanning Trees, Bachelor Thesis, Spring 2011.

Row sums of A339072.

Cf. A000088, A001349, A002218, A006289, A007112, A338511.

More terms from Ronald C. Read.

A010355 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 16, 42, 111, 331, 1094, 3829, 14380, 57069, 237188, 1027929, 4622588, 21494274, 103077677, 508743475, 2579847563, 13422868110, 71570635306, 390670937143

Offset: 1

N. J. A. Sloane

Original name: Single-edge stars with n edges.

			From _Andrew Howroyd_, Nov 23 2020: (Start)
The a(1) = 1 graph is the single edge (K_2 = P_2).
The a(3) = 1 graph is the triangle (K_3).
The a(4) = 1 graph is the square (C_4).
The a(5) = 2 graphs are the cycle C_5 and a cycle of 4 nodes with one diagonal added.
(End)

George A. Baker Jr. and John M. Kincaid, The continuous-spin Ising model, g0:phi4:d field theory and the renormalization group. J. Statist. Phys. 24 (1981), no. 3, 469-528.
Brendan McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation, 60 (2014), pp. 94-112.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.
M. F. Sykes, D. S. McKenzie, and B. R. Heap, Specific heat of a three-dimensional Ising ferromagnet above the Curie temperature. III. The star cluster expansion, J. Phys. A: Math. Nucl. Gen., 7 (1974), 1576.

Row sums of A339070 and A010356.
Column sums of A339071.
Cf. A002218, A010357, A322139, A338511.

a(11)-a(12) from Andrey Zabolotskiy, Oct 03 2017
Name changed by Andrew Howroyd, Nov 23 2020
a(13)-a(18) added using data from Robinson's tables by Andrew Howroyd, Nov 23 2020
a(19)-a(22) from Hugo Pfoertner using program geng from nauty, Dec 04 2020
a(23)-a(24) from Hugo Pfoertner, Dec 07 2020
a(25) from Hugo Pfoertner, Jan 04 2021

A002880 Number of 3-connected nets with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 9, 11, 37, 79, 249, 671, 2182, 6692, 22131, 72405, 243806, 822788, 2815119, 9679205, 33551192, 116900081, 409675567, 1442454215, 5102542680, 18124571838, 64634480340, 231334873091, 830828150081, 2993489821771

Offset: 6

N. J. A. Sloane

Also, the number of 3-connected quadrangulations without separating 4-cycles (up to orientation) with n faces. - Andrey Zabolotskiy, Sep 20 2019

			G.f. = x^6 + x^8 + x^9 + 2*x^10 + 2*x^11 + 9*x^12 + 11*x^13 + 37*x^14 + ...

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).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
J. A. D. Cameron, Searching for Squared Squares, USYD Master of Science Thesis (1976). Rare Books & Special Collections Fisher Library, Sydney University.
J. A. D. Cameron, Table 7.2 - listing of tri-connected planar graphs by edge, from the thesis - this was the first count of order 20 (22131). Photo by Stuart E Anderson.
G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
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]
CombOS - Combinatorial Object Server, generate planar graphs
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
N. D. Kazarinoff and R. Weitzenkamp, Squaring rectangles and squares, Amer. Math. Monthly, 80 (1973), 877-888.

Cf. A002840, A002841, A113201, A078666, A007022, A113205, A338511.

A338593 Number of unlabeled connected nonplanar graphs with n edges with degree >= 3 at each node.

Original entry on oeis.org

1, 2, 3, 10, 30, 100, 371, 1419, 5764, 24482, 107583, 487647, 2271488, 10847623

Offset: 9

Hugo Pfoertner, Nov 21 2020

First differs from A338583 for n = 13. All unlabeled nonplanar graphs with n <= 12 edges and degree >= 3 at each node are 3-connected. For this reason the illustrations of the graphs are identical up to n = 12. The first differences for n = 13 and n = 14 are shown in the illustrations of A338584.

Hugo Pfoertner, Illustrations of terms a(9) - a(12), from A338583.
Hugo Pfoertner, Illustration of terms a(13) and a(14), showing graphs counted in this sequence, but not in A338583 (illustrations from A338584).
Hugo Pfoertner, Unlabeled nonplanar graphs with vertex degree >=3 for n<=19 edges, list in PARI-readable format.

Cf. A007112, A002840, A123545, A338511, A338583, A338584, A338594, A338604.

\\ It is assumed that the a338593.gp file (from the linked zip archive) has been read before, i.e., \r [path]a338593.gp
for(k=9,#EdgeDataNonplanarDegge3,print1(#EdgeDataNonplanarDegge3[k],", "));
\\ printing of the edge lists of the graphs for n <= 11
print(EdgeDataNonplanarDegge3[9..11])

a(n) = A338604(n) - A338594(n).

A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3n/2) <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 2, 1, 1, 3, 14, 25, 31, 28, 17, 9, 5, 2, 1, 1, 4, 24, 101, 254, 413, 475, 426, 306, 187, 103, 52, 23, 11, 5, 2, 1, 1, 19, 204, 1068, 3348, 7152, 11199, 13683, 13604, 11374, 8203, 5216, 2963, 1536, 737, 333, 144, 62, 25, 11, 5, 2, 1, 1

Offset: 4

Andrew Howroyd, Nov 24 2020

			Triangle begins:
===========================================================
n/k | 6  7  8   9  10 11  12  13  14  15  16 17 18 19 20 21
----+------------------------------------------------------
  4 | 1;
  5 |       1,  1,  1;
  6 |           2,  3, 4,  4,  2,  1,  1;
  7 |                  3, 14, 25, 31, 28, 17, 9, 5, 2, 1, 1;
  8 |                      4, 24 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 4..732 (rows n=4..18, extracted from Robinson's tables)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table VI, pages 19-27.

Row sums are A006290.
Column sums are A338511.
Cf. A054924, A123542, A339071.

A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 15, 121, 2159, 68715, 3952378, 389968005, 65161587084

Offset: 1

Eric W. Weisstein, May 08 2000

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Combinatorial data.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=3 of A263296.
Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A241703, A241704, A241705.
Cf. A338511, A338583, A338593, A338594, A338604, A339072.

a(8), a(9), a(10) from the Encyclopedia of Finite Graphs by Travis Hoppe and Anna Petrone, Apr 22 2014
a(11) by Jens M. Schmidt, Feb 18 2019
a(12) from Jens M. Schmidt's web page, Jan 10 2021

A338594 Number of unlabeled connected planar graphs with n edges with degree >= 3 at each node.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 17, 37, 98, 275, 797, 2414, 7613, 24510, 80721, 270018, 915034

Offset: 6

Hugo Pfoertner, Nov 21 2020

			a(6) = 1: the 3-connected edge graph of the tetrahedron;
a(7) = 0: no connected planar graph with degree >=3 at each node exists;
a(8) = 1: the 3-connected 5-wheel graph, edge graph of 4-sided pyramid;
a(9)-a(11): see linked illustrations.

Hugo Pfoertner, Illustrations of terms a(9) - a(11).

Cf. A007112, A002840, A123545, A338511, A338583, A338584, A338593, A338604.

a(n) = A338604(n) - A338593(n).

A338604 Number of unlabeled connected graphs with n edges with degree >= 3 at each node.

Original entry on oeis.org

1, 0, 1, 3, 5, 9, 27, 67, 198, 646, 2216, 8178, 32095, 132093, 568368, 2541506, 11762657, 56183633, 276288402, 1396172601, 7238931364

Offset: 6

Hugo Pfoertner, Nov 21 2020

			a(10)=5:
There are 5 graphs with 10 edges and degree >=3 at all nodes (see table in A123545):
The complete graph on 5 nodes, given by the edge list
[[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],
and 4 graphs on 6 nodes:
  [[1,3],[1,5],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,5],[4,6]],
  [[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4],[3,5],[3,6],[4,6]],
  [[1,3],[1,4],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,6],[5,6]],
  [[1,3],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,6]].
The first one has degree 3 or 4 at all nodes, but becomes disconnected by removing nodes 5 and 6 and their incident edges. It is therefore not 3-connected.
    .--5--.
   /  / \  \
  1--3   4--2
   \  \ /  /
    .--6--.
.
The complete graph on 5 nodes and the last 3 graphs with 6 nodes are all 3-connected. Thus A338511(10)=4, and by inclusion of the graph shown above a(10)=5.

Cf. A007112, A123545, A338511, A338593, A338594.

cp's OEIS Frontend

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

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A002840 Number of polyhedral graphs with n edges.

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A006290 Number of 3-connected graphs with n nodes.

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A010355 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n edges.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A002880 Number of 3-connected nets with n edges.

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Comments

Examples

References

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Crossrefs

A338593 Number of unlabeled connected nonplanar graphs with n edges with degree >= 3 at each node.

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Author

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Programs

PARI

Formula

A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n*(n-1)/2).

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A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

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Author

Keywords

Links

Crossrefs

Extensions

A338594 Number of unlabeled connected planar graphs with n edges with degree >= 3 at each node.

Views

Author

Keywords

Examples

A339072 Triangle read by rows: T(n,k) is the number of unlabeled simple 3-connected graphs with n nodes and k edges (n >= 4, ceiling(3n/2) <= k <= n(n-1)/2).