cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A210689 Erroneous version of A129416.

Original entry on oeis.org

1, 3, 9, 32, 137
Offset: 1

Views

Author

N. J. A. Sloane, Mar 30 2012

Keywords

References

  • Cadogan, C. C. Graph-lattices and the enumeration of linear graphs. 1970 Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1970) pp. 61--86 Louisiana State Univ., Baton Rouge, La. MR0269553 (42 #4448).

A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

Original entry on oeis.org

1, 2, 6, 20, 91, 509, 3608, 31856, 340416, 4269971, 61133757, 978098997, 17228295555, 330552900516, 6853905618223, 152626436936272, 3631575281503404, 91928898608055819, 2466448432564961852, 69907637101781318907
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the number of isomorphism classes of connected 3-regular simple graphs of order 2n with possibly loops. - Nico Van Cleemput, Jun 04 2014
There are no graphs of order 2n+1 satisfying the condition above. - Natan Arie Consigli, Dec 20 2019

Examples

			From _Natan Arie Consigli_, Dec 20 2019: (Start)
a(1) = 1: with two nodes the only viable option is the triple edged path multigraph.
a(2) = 4: with four nodes we have two cases: the tetrahedral graph and the square graph with single and double edges on opposite sides.
(End)

References

  • A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92 [gives incorrect a(6)].
  • CRC Handbook of Combinatorial Designs, 1996, p. 651 [or: 2006, table 4.40].

Links

Crossrefs

Column k=3 of A328682 (table of k-regular n-node multigraphs).
Cf. A129416, A005967 (loops allowed), A129417, A129419, A129421, A129423, A129425, A002851 (no multiedges).

Programs

  • nauty
    for n in {1..10}; do geng -cqD3 $[2*$n] | multig -ur3; done # Sean A. Irvine, Sep 24 2015

Formula

Inverse Euler transform of A129416. - Andrew Howroyd, Mar 19 2020

Extensions

More terms from Brendan McKay, Apr 15 2007
a(13)-a(20) from Andrew Howroyd, Mar 19 2020

A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

1, 2, 8, 31, 140, 722, 4439, 32654, 289519, 3054067, 37584620, 527968286, 8308434931, 144345554051, 2738280739075, 56245013793246, 1242596591479816, 29366532494796900, 739033832149588904, 19726887762569763453
Offset: 0

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

a(1)..a(11) computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

References

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

Links

Crossrefs

Column k=3 of A167625.
Cf. A005967 (connected, inv. Euler trans.), A129416, A129429, A129431, A129433, A129435, A129437, A005638.

Programs

  • Sage
    h = SymmetricFunctions(QQ).homogeneous()
def A129427(n):
    X = h([2*n]).plethysm(h([3]))
    Y = h([3*n]).plethysm(h([2]))
    return X.scalar(Y)
# Bruce Westbury, Aug 16 2013

Formula

a(n)=N\{S_{2n}[S_3] * S_{3n}[S_2]\}. - Jason Kimberley, Sep 17 2009

Extensions

Using equation (5.8) of Read 1959, new terms a(12) and a(13) were computed in MAGMA by Jason Kimberley, Sep 17 2009
Further terms a(14)-a(16) also computed by Jason Kimberley, announced Nov 09 2009
Formula corrected from n vertices to 2n vertices by Jason Kimberley, Nov 09 2009
Added a(0). - N. J. A. Sloane, Aug 26 2013
a(17)-a(19) from Sean A. Irvine, Oct 29 2016

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 3, 2, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 0, 1, 1, 5, 7, 9, 4, 1, 1, 1, 0, 1, 0, 7, 0, 24, 0, 7, 0, 1, 1, 0, 1, 1, 8, 16, 54, 60, 32, 8, 1, 1, 1, 0, 1, 0, 10, 0, 128, 0, 240, 0, 12, 0, 1, 1, 0, 1, 1, 12, 37, 271, 955, 1753, 930, 135, 14, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Mar 15 2020

Keywords

Comments

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

Examples

			Array begins:
=================================================
n\k | 0 1 2  3   4    5      6     7        8
----+--------------------------------------------
  0 | 1 1 1  1   1    1      1     1        1 ...
  1 | 1 0 0  0   0    0      0     0        0 ...
  2 | 1 1 1  1   1    1      1     1        1 ...
  3 | 1 0 1  0   1    0      1     0        1 ...
  4 | 1 1 2  3   4    5      7     8       10 ...
  5 | 1 0 2  0   7    0     16     0       37 ...
  6 | 1 1 4  9  24   54    128   271      582 ...
  7 | 1 0 4  0  60    0    955     0    12511 ...
  8 | 1 1 7 32 240 1753  13467 90913   543779 ...
  9 | 1 0 8  0 930    0 253373     0 35255015 ...
  ...

Links

Crossrefs

Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.
Row n=4 is A001399.
Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

A129426 Number of isomorphism classes of 8-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 10, 37, 582, 12511, 543779, 35255015, 3230979297, 397550237967, 63834143947661, 13080849749829233, 3358751856150607392, 1063851391062768324862, 410060430118305494628648, 190065946515113295597969794, 104826174445642584491349328181
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=8 of A333330.
Cf. A129425, A129437, A129416, A129418, A129420, A129422, A129424.

Formula

Euler transform of A129425. - Andrew Howroyd, Mar 17 2020

Extensions

a(1)=0 prepended and a(12)-a(18) from Andrew Howroyd, Mar 17 2020

A129418 Number of isomorphism classes of 4-regular loopless multigraphs of order n.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 24, 60, 240, 930, 4701, 26637, 178569, 1339529, 11187064, 101871881, 1002594996, 10574095327, 118850827173, 1417140114336, 17860018997346, 237160827107408, 3309078044759285, 48396906463199522, 740331404753448181, 11821525310570525197
Offset: 0

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/
Also number of carbon allotropes satisfying the octet rule, excluding stereoisomers. - Natan Arie Consigli, Jun 06 2017

Crossrefs

Column k=4 of A333330.
Cf. A129417, A129429, A129416, A129420, A129422, A129424, A129426.

Programs

Formula

Euler transform of A129417. - Andrew Howroyd, Mar 14 2020

Extensions

a(0)-a(1) by Natan Arie Consigli, Jun 06 2017
a(18)-a(25) from Andrew Howroyd, Mar 17 2020

A129420 Number of isomorphism classes of 5-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 5, 54, 1753, 189341, 46935710, 20494522535, 14041749098602, 14155266802426836, 20061744131278672638, 38587417589460488631726, 97900485588988429336271590, 320012505326477694925887757141, 1321269556386383657509085883067690, 6775074159053505093089897813890701467
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=5 of A333330.
Cf. A129419, A129431, A129416, A129418, A129422, A129424, A129426.

Formula

Euler transform of A129419. - Andrew Howroyd, Mar 17 2020

Extensions

a(8)-a(15) from Andrew Howroyd, Mar 21 2020

A129422 Number of isomorphism classes of 6-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 7, 16, 128, 955, 13467, 253373, 6466074, 205749149, 7943313377, 363853255012, 19485170158346, 1205488841884007, 85308028236495340, 6846462326434510551, 618498122199399056707, 62478078728492272712838, 7015617595855429187696753
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=6 of A333330.
Cf. A129421, A129433, A129416, A129418, A129420, A129424, A129426.

Formula

Euler transform of A129421. - Andrew Howroyd, Mar 14 2020

Extensions

a(1)=0 prepended and a(14)-a(20) from Andrew Howroyd, Mar 17 2020

A129424 Number of isomorphism classes of 7-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 8, 271, 90913, 165134083, 861889423083, 10352781874872853, 253226977418079364874, 11542716741789258789499297, 915461016322959135074642849828
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=7 of A333330.
Cf. A129423, A129435, A129416, A129418, A129420, A129422, A129426.

Formula

Euler transform of A129423. - Andrew Howroyd, Mar 21 2020

Extensions

a(7)-a(10) from Andrew Howroyd, Mar 21 2020
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.