A002840 -id:A002840 - cp's OEIS frontend

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

1, 0, 1, 1, 2, 4, 7, 16, 41, 108, 320, 1042, 3575, 13064, 49938, 197729, 805991, 3363084, 14302891, 61813285, 270805177, 1200460492, 5376709415, 24302430375, 110745093999, 508380790741

Offset: 1

Andrew Howroyd, May 04 2021

Terms may be computed using the tools geng and planarg in nauty.

Brendan McKay and Adolfo Piperno, nauty and Traces

Row sums of A343870.
Column sums of A049336(n > 1).
Cf. A002840 (3-connected), A010355, A021103, A046091, A289471, A291841.

# 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, Jun 05 2023

a(21)-a(26) added by Georg Grasegger, Jun 05 2023

A005645 Number of sensed 3-connected planar maps with n edges.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 15, 32, 89, 266, 797, 2496, 8012, 26028, 85888, 286608, 965216, 3278776, 11221548, 38665192, 134050521, 467382224, 1638080277, 5768886048, 20407622631, 72494277840, 258527335373, 925322077852, 3323258053528, 11973883092034, 43273374700200, 156836969693756, 569967330200576, 2076647113454878, 7584534277720818, 27764845224462192, 101862027752012402, 374484866509396780, 1379489908513460150

Offset: 6

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 6..50
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
T. R. S. Walsh, Counting nonisomorphic three-connected planar maps, J. Combin. Theory Ser. B 32 (1982), no. 1, 33-44.

Cf. A002840 (unsensed), A239893.

a(n) = Sum_{k=4..n-2} A239893(k, n+2-k). - Andrew Howroyd, Mar 27 2021

More terms and b-file added by N. J. A. Sloane, May 08 2012

A378076 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 16, 37, 100, 293, 888, 2822, 9305, 31274, 106972, 370828, 1298058, 4582413, 16289759, 58259600, 209465186

Offset: 6

Andrew Howroyd, Nov 15 2024

Homeomorphically irreducible means each vertex has a degree of at least 3.

Antidiagonal sums of A378075.

Cf. A002840, A006407, A187928.

a(n) = A002840(n) + A187928(n).

a(n) = Sum_{k=4..n-2} A378075(k, n+2-k).

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

A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3807, 11549, 34843, 104459, 311317, 928719, 2776247, 8320757, 24967341, 74985337

Offset: 0

Hugo Pfoertner and Rainer Rosenthal, Feb 14 2021

			a(10) = 3807, whereas A337517(10) = 3823. The difference of 16 resistances results from the 15 terms of A338601/A338602 and the resistance 34/27 not representable by a planar network of 10 resistors, whereas it (but not 27/34) can be represented by a nonplanar network of 10 resistors.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles (SPSR).
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.

Cf. A002840, A048211, A174283, A180414, A337516, A337517, A338511, A338601, A338602, A340921.

Cf. A113881, A219158.

Replace the list of 3-connected graphs corresponding to A338511 by the list of planar graphs provided in A002840 in Andrew Howroyd's PARI program linked in A180414.

a(n) = A337517(n) for n <= 9, a(n) < A337517(n) for n >= 10.

a(19) from Hugo Pfoertner, Mar 15 2021

A343871 Number of labeled 3-connected planar graphs with n edges.

Original entry on oeis.org

1, 0, 15, 70, 432, 4320, 30855, 294840, 2883240, 28175952, 310690800, 3458941920, 40459730640, 499638948480, 6324655705200, 83653192972800, 1145266802114400, 16145338385736000, 235579813593453000, 3535776409508703360, 54571687068401395200, 866268656574795936000

Offset: 6

Andrew Howroyd, May 05 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..200
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

Cf. A000287, A002840 (unlabeled case), A096330, A290326, A291841, A338414.

Q(n,k) = { \\ c-nets with n-edges, k-vertices (see A290326)
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
a(n)={sum(k=2+(n+2)\3, 2*n\3, k!*Q(n,k))/(4*n)} \\ Andrew Howroyd, May 05 2021

a(n) = Sum_{k=2+floor((n+2)/3)..floor(2*n/3)} k!*A290326(n-k+1, k-1)/(4*n).

A289470 Number of strictly 2-connected graphs with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 15, 39, 107, 324, 1072, 3778, 14228, 56568, 235449, 1021381, 4596328, 21383982, 102594132, 506544749, 2569520447, 13372902590, 71322154244, 389402949706

Offset: 1

Ed Pegg Jr, Jul 06 2017

Cf. A002218, A002840, A010355, A289471, A338511.

a(n) = A010355(n) - A338511(n). - Andrew Howroyd, May 03 2021

a(12)-a(13) corrected and a(14)-a(25) from Andrew Howroyd, May 03 2021

A355638 Number of polyhedra (3-polytopes) of graph radius 1 on n edges.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 4, 5, 7, 10, 16, 27, 42, 67, 116, 187, 329, 570, 970, 1723, 3021, 5338, 9563, 16981, 30517, 54913, 98847, 179119, 324333, 589059, 1072997, 1955207, 3573129, 6538088

Offset: 6

Riccardo Maffucci, Jul 11 2022

Data was gathered with the help of Scientific IT & Application Support (SCITAS) High Performance Computing (HPC) for the EPFL community.

			For n=6 there is only the tetrahedron, n=8 the square pyramid, n=9 the triangular bipyramid,...

R. W. Maffucci, On unigraphic 3-polytopes of radius one, arXiv:2207.02040 [math.CO], 2022.

Cf. A002840.

Needs["IGraphM`"]
ra[8]:={Square pyramid}
ra[q]=opb[ra[q-1]]
opb[setg_] :=
Prepend[DeleteDuplicatesBy[
   Flatten[Table[
     EdgeAdd[g, UndirectedEdge[x[[1]], x[[2]]],
      GraphLayout -> "TutteEmbedding"], {g, setg}, {x,
      Flatten[Table[
        Complement[Subsets[i, {2}],
         Table[{i[[j]], i[[j + 1]]}, {j, Length[i] - 1}], {{i[[1]],
           i[[-1]]}}], {i, Select[IGFaces[g], Length[#] > 3 &]}],
       1]}]], CanonicalGraph],
  If[OddQ[EdgeCount[setg[[1]]]],
   WheelGraph[EdgeCount[setg[[1]]]/2 + 3/2,
    GraphLayout -> "TutteEmbedding", ImageSize -> 25], Nothing]]

cp's OEIS Frontend

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

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nauty

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A005645 Number of sensed 3-connected planar maps with n edges.

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A378076 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n edges.

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

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A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

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PARI

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Extensions

A343871 Number of labeled 3-connected planar graphs with n edges.

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Programs

PARI

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A289470 Number of strictly 2-connected graphs with n edges.

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Keywords

Crossrefs

Formula

Extensions

A355638 Number of polyhedra (3-polytopes) of graph radius 1 on n edges.

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Author

Keywords

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Programs

Mathematica