A005638 -id:A005638 - cp's OEIS frontend

A185336 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 6.

1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7574, 181227, 4624502, 122090545, 3328929960, 93990692632, 2754222605808

Offset: 0

Jason Kimberley, Jan 28 2012

The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 6: A014374 (connected), A185236 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 6: A185326 (k=2), this sequence (k=3).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), A185335 (g=5), this sequence (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

A014374 = Cases[Import["https://oeis.org/A014374/b014374.txt", "Table"], {, }][[All, 2]];
etr[f_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d f[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
a = etr[A014374[[# + 1]]&];
a /@ Range[0, Length[A014374] - 1] (* Jean-François Alcover, Dec 04 2019 *)

Euler transformation of A014374.

a(18) from A014374 from Jean-François Alcover, Dec 04 2019

A361408 Number of connected cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Original entry on oeis.org

0, 1, 5, 31, 248, 2382, 27233, 359800, 5364193, 88622485, 1602171855, 31410476113, 663240471075, 15001046054183, 361775504849332, 9266474332849318, 251217335356943672, 7186461542458525108, 216332059500870350414, 6835872042063656823802

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=2 of A321304.

Cf. A002851, A005638, A361407, A361411.

G.f.: B(x)/C(x) - (D(x) + D(x^2))/2 where B(x), C(x) and D(x) are the g.f.s of A361411, A005638 and A361407, respectively.

A361410 Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.

Original entry on oeis.org

0, 1, 2, 11, 68, 510, 4712, 51877, 664520, 9662968, 156490473, 2783955994, 53863486240, 1124886942314, 25206326633070, 603048386506505, 15339533779133582, 413338072569232815, 11760801736217845686, 352342902996056683824

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=1 of A361361.

Cf. A005638, A361407, A361411.

A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Original entry on oeis.org

0, 1, 5, 33, 257, 2443, 27682, 363759, 5405697, 89134360, 1609418390, 31525697245, 665263778962, 15039817276939, 362579178545598, 9284375250749758, 251643492565059981, 7197256536139662143, 216621907269166632361, 6844093745422473471562

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=2 of A361361.

Cf. A005638, A361408, A361410.

A003175 Almost certainly an erroneous version of A129427.

Original entry on oeis.org

1, 2, 8, 31, 139, 724

Offset: 0

N. J. A. Sloane, Jul 02 2015

dead

P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.

Cf. A129427, A005638.

A184325 The number of disconnected 2k-regular simple graphs on 4k+5 vertices.

Original entry on oeis.org

1, 3, 8, 25, 100, 550, 4224, 42135, 516383, 7373984, 118573680, 2103205868, 40634185593, 847871397697, 18987149095396, 454032821689310, 11544329612486760, 310964453836199398, 8845303172513782781

Offset: 0

Jason Kimberley, Jan 11 2011

nonn

			The a(0)=1 graph is 5K_1. The a(1)=3 graphs are 3C_3, C_3+C_6, and C_4+C_5.

This sequence is the (even indices of the) fourth highest diagonal of D=A068933: that is a(n) = D(4k+5, 2k).

Cf. A184324(k) = D(2k+4, k) and A184326(k) = D(2k+6, k).

a(0)=1. For n > 0, a(n) = A051031(2k+4,3) + A051031(2k+3,2) = A005638(k+2) + A008483(2k+3).

Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(4k+5,2k) = C(2k+1, 2k) C(2k+4,2k) + C(2k+2,2k) C(2k+3,2k), from the disconnected Euler transform. For n > 1, D(2k+1,2k) = D(2k+2,2k) = D(2k+3,2k) = D(2k+4,2k) = 0. Therefore, a(n) = E(2k+1, 2k) E(2k+4,2k) + E(2k+2,2k) E(2k+3,2k) = E(2k+1,0) E(2k+4,3) + E(2k+2,1) E(2k+3,2). Note that E(2k+1,0) = E(2k+2,1) = 1. Checking a(1) = E(6,3) + E(5,2), QED.

A184326 The number of disconnected k-regular simple graphs on 2k+6 vertices.

Original entry on oeis.org

1, 1, 4, 9, 25, 66, 297, 1562, 10901, 88238, 806174, 8037887, 86228020, 985884104, 11946634677, 152808994328, 2056701656260

Offset: 0

Jason Kimberley, Jan 15 2011

			The a(0)=1 graph is 6K_1. The a(1)=1 graph is 4K_2. The a(2)=4 graphs are 2C_3+C_4, 2C_5, C_4+C_6, and C_3+C_7.

This sequence is the fifth highest diagonal of D=A068933: that is a(n)=D(2k+6, k).

Cf. A184324(k) = D(2k+4, k) and A184325(k) = D(4k+5, 2k).

a(0)=1, a(1)=1, a(2)=4, a(3)=9. For n>3, a(n) = A033301(k+5) + ((k+1)mod 2)*A005638(k div 2 + 2) + A000217(A008483(k+3)).

Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+6,k) = C(k+1,k)C(k+5,k) + C(k+2,k)C(k+4,k) + A000217(C(k+3,k)), from the disconnected Euler transform. Notice that D(k+i,k)=0 provided k+i < 2k+2; that is k > i-2. So if i <= 5 and k > 3, then D(k+i,k)=0. Hence for k > 3, a(n) = E(k+1,k)E(k+5,k) + E(k+2,k)E(k+4,k) + A000217(E(k+3,k)) = E(k+1,0)E(k+5,4) + E(k+2,1)E(k+4,3) + A000217(E(k+3,2)). We have E(k+1,0)=1, and E(k+2,1)=(k+1)mod 2. For even k, E(k+4,3)=A005638(k div 2 + 2); for odd k, E(k+2,1)=0. QED.

A185330 Irregular triangle E(n,g) counting not necessarily connected 3-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 2, 1, 6, 2, 21, 6, 1, 94, 23, 2, 540, 112, 9, 1, 4207, 801, 49, 1, 42110, 7840, 455, 5, 516344, 97723, 5784, 32, 7373924, 1436873, 90940, 385, 118573592, 23791155, 1620491, 7574, 1, 2103205738, 432878091, 31478651, 181227, 3, 40634185402

Offset: 2

Jason Kimberley, Oct 18 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A000066(g).

			1;
2, 1;
6, 2;
21, 6, 1;
94, 23, 2;
540, 112, 9, 1;
4207, 801, 49, 1;
42110, 7840, 455, 5;
516344, 97723, 5784, 32;
7373924, 1436873, 90940, 385;
118573592, 23791155, 1620491, 7574, 1;
2103205738, 432878091, 31478651, 181227, 3;
40634185402, 8544173926, 656784488, 4624502, 21;
847871397424, 181519645163, 14621878339, 122090545, 546, 1;
18987149095005, 4127569521160, 345975756388, 3328929960, 30368, 0;

Jason Kimberley, Table of i, a(i)=E(n,g) for i = 2..60 (n = 2..16)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Cf. A005638, A185334, A185304.

A275744 Triangle read by rows: Number of unlabeled cubic graphs with 2n nodes and k components.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 5, 1, 0, 0, 19, 2, 0, 0, 0, 85, 8, 1, 0, 0, 0, 509, 29, 2, 0, 0, 0, 0, 4060, 138, 8, 1, 0, 0, 0, 0, 41301, 774, 33, 2, 0, 0, 0, 0, 0, 510489, 5693, 153, 8, 1, 0, 0, 0, 0, 0, 7319447, 53581, 861, 33, 2, 0, 0, 0, 0, 0, 0, 117940535, 626717, 6173, 158, 8, 1

Offset: 1

R. J. Mathar, Aug 07 2016

Multiset transformation of A002851.

			The triangle starts
      0;
      1       0;
      2       0       0;
      5       1       0       0;
     19       2       0       0       0;
     85       8       1       0       0       0;
    509      29       2       0       0       0       0;
   4060     138       8       1       0       0       0       0;
  41301     774      33       2       0       0       0       0       0;
.510489    5693     153       8       1       0       0       0       0       0;
...

Index entries for triangles generated by the Multiset Transformation

Cf. A005638 (row sums).

T(n,1) = A002851(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} (1-y*x^j)^(-A002851(j)). - Alois P. Heinz, Apr 13 2017

A333163 Number of cubic graphs on n unlabeled nodes with half-edges.

Original entry on oeis.org

1, 0, 0, 1, 3, 4, 12, 24, 70, 172, 525, 1530, 5078, 16994, 61456, 228898, 895910, 3617148, 15130833, 65084088, 287828488, 1304327221, 6050218591, 28675928883, 138730847262, 684300453848, 3438439910436, 17585597712632, 91479580896616, 483699938173293, 2598090378779507

Offset: 0

Andrew Howroyd, Mar 11 2020

nonn

A half-edge is like a loop except it only adds 1 to the degree of its vertex.

a(n) is the number of simple graphs on n unlabeled nodes with every node having degree 2 or 3.

Column k=3 of A333161.

Cf. A005638, A110040, A186417.

Previous Showing 21-30 of 33 results. Next

cp's OEIS Frontend

A185336 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A361408 Number of connected cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Views

Author

Keywords

Crossrefs

Formula

A361410 Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.

Views

Author

Keywords

Crossrefs

A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Views

Author

Keywords

Crossrefs

A003175 Almost certainly an erroneous version of A129427.

Views

Author

Keywords

References

Links

Crossrefs

A184325 The number of disconnected 2k-regular simple graphs on 4k+5 vertices.

Views

Author

Keywords

Examples

Crossrefs

Formula

A184326 The number of disconnected k-regular simple graphs on 2k+6 vertices.

Views

Author

Keywords

Examples

Crossrefs

Formula

A185330 Irregular triangle E(n,g) counting not necessarily connected 3-regular simple graphs on 2n vertices with girth at least g.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A275744 Triangle read by rows: Number of unlabeled cubic graphs with 2n nodes and k components.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A333163 Number of cubic graphs on n unlabeled nodes with half-edges.

Views

Author

Keywords

Comments

Crossrefs