A123545 -id:A123545 - 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
```
(* See link. *)
```

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

A007112 Number of connected unlabeled graphs with n nodes and degree >= 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 19, 150, 2589, 84242, 5203110, 577076528, 113373005661, 39618075274687, 24916469690421103, 28563628406172313565, 60366734880279992689057, 237406976195431811992516687, 1750330442243433249473074447457, 24333391254830431938389159467111420, 640811881795531742198398521689947259687

Offset: 1

N. J. A. Sloane

R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Table of n, a(n) for n = 1..28
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Minimum Vertex Degree.

Row sums of A123545.

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).

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.

A338584 Number of unlabeled nonplanar connected graphs with n edges with minimum degree 3 at each node that are not 3-connected.

Original entry on oeis.org

1, 6, 28, 128, 558, 2421, 10675, 47810, 217572, 1006211

Offset: 13

Hugo Pfoertner, Nov 21 2020

Hugo Pfoertner, Illustration of terms a(13) and a(14).

Cf. A006290, A007112, A123545, A338583, A338593, A338594, A338604.

a(n) = A338593(n) - A338583(n).

A123546 Triangle read by rows: T(n,k) = number of unlabeled graphs on n nodes with degree >= 3 at each node (n >= 1, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 5, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 18, 30, 34, 29, 17, 9, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 35, 136, 309, 465, 505, 438, 310, 188, 103, 52, 23

Offset: 0

N. J. A. Sloane, Nov 14 2006

			Triangle begins:
n = 0
k = 0 : 0
************************* total (n = 0) = 0
n = 1
k = 0 : 0
************************* total (n = 1) = 0
n = 2
k = 0 : 0
k = 1 : 0
************************* total (n = 2) = 0
n = 3
k = 0 : 0
k = 1 : 0
k = 2 : 0
k = 3 : 0
************************* total (n = 3) = 0
n = 4
k = 0 : 0
k = 1 : 0
k = 2 : 0
k = 3 : 0
k = 4 : 0
k = 5 : 0
k = 6 : 1
************************* total (n = 4) = 1
n = 5
k = 0 : 0
k = 1 : 0
k = 2 : 0
k = 3 : 0
k = 4 : 0
k = 5 : 0
k = 6 : 0
k = 7 : 0
k = 8 : 1
k = 9 : 1
k = 10 : 1
************************* total (n = 5) = 3

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

R. W. Robinson, Rows 0 through 14, flattened

Row sums give A007111. Cf. A007112, A123545.

A339069 Triangle read by rows: T(n,k) is the number of unlabeled simple series-reduced 2-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, 4, 5, 4, 2, 1, 1, 4, 17, 30, 34, 29, 17, 9, 5, 2, 1, 1, 5, 33, 133, 307, 464, 505, 438, 310, 188, 103, 52, 23, 11, 5, 2, 1, 1, 25, 277, 1352, 3953, 7939, 11897, 14131, 13827, 11465, 8235, 5226, 2966, 1537, 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,  4, 5,  4,  2,  1,  1;
  7 |                  4, 17, 30, 34, 29, 17, 9, 5, 2, 1, 1;
  8 |                      5, 33 ...
  ...

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 V, pages 10-18.
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.

Row sums are A006289.
Column sums are A339068.
Cf. A123545, A123546, A339071, A339072.

cp's OEIS Frontend

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

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A007112 Number of connected unlabeled graphs with n nodes and degree >= 3.

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

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

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

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A338584 Number of unlabeled nonplanar connected graphs with n edges with minimum degree 3 at each node that are not 3-connected.

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A123546 Triangle read by rows: T(n,k) = number of unlabeled graphs on n nodes with degree >= 3 at each node (n >= 1, 0 <= k <= n(n-1)/2).

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A339069 Triangle read by rows: T(n,k) is the number of unlabeled simple series-reduced 2-connected graphs with n nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n*(n-1)/2).

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A339069 Triangle read by rows: T(n,k) is the number of unlabeled simple series-reduced 2-connected graphs with n nodes and k edges (n >= 4, ceiling(3n/2) <= k <= n(n-1)/2).