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 1-6 of 6 results.

A014381 Number of connected regular graphs of degree 9 with 2n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 9, 88193, 113314233813, 281341168330848874, 1251392240942040452186674, 9854603833337765095207342173991, 134283276101750327256393048776114352985
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Since the nontrivial 9-regular graph with the least number of vertices is K_10, there are no disconnected 9-regular graphs with less than 20 vertices. Thus for n<20 this sequence also gives the number of all 9-regular graphs on 2n vertices. - Jason Kimberley, Sep 25 2009

Examples

			The null graph on 0 vertices is vacuously connected and 9-regular; since it is acyclic, it has infinite girth. - _Jason Kimberley_, Feb 10 2011

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 648.
  • I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.

Links

Crossrefs

Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), this sequence (k=9), A014382 (k=10), A014384 (k=11).
9-regular simple graphs: this sequence (connected), A185293 (disconnected).
Connected 9-regular simple graphs with girth at least g: this sequence (g=3), A181170 (g=4).
Connected 9-regular simple graphs with girth exactly g: A184993 (g=3).

Formula

a(n) = A184993(n) + A181170(n).

Extensions

a(8) appended using the symmetry of A051031 by Jason Kimberley, Sep 25 2009
a(9)-a(10) from Andrew Howroyd, Mar 13 2020
a(10) corrected and a(11)-a(12) from Andrew Howroyd, May 19 2020

A184943 Number of connected 4-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 5, 16, 57, 263, 1532, 10747, 87948, 803885, 8020590, 86027734, 983417704, 11913817317, 152352034707, 2050055948375, 28951137255862, 428085461764471
Offset: 0

Views

Author

Jason Kimberley, Jan 25 2011

Keywords

Examples

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(5)=1 complete graph on 5 vertices is 4-regular; it has 10 edges and 10 triangles.

Links

Crossrefs

4-regular simple graphs with girth exactly 3: this sequence (connected), A185043 (disconnected), A185143 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 3: A006923 (k=3), this sequence (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: this sequence (g=3), A184944 (g=4), A184945 (g=5).

Programs

  • Mathematica
    A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A006820 = A@006820; A033886 = A@033886;
a[n_] := A006820[[n + 1]] - A033886[[n + 1]];
a /@ Range[0, 22] (* Jean-François Alcover, Jan 27 2020 *)

Formula

a(n) = A006820(n) - A033886(n).

Extensions

Term a(22) corrected and a(23) appended, due to the correction and extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 13 2020

A181170 Number of connected 9-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 14
Offset: 0

Views

Author

Jason Kimberley, last week of Jan 2011

Keywords

Comments

a(11)=14 was computed by the author using GENREG at U. Ncle. over 615 processor days during Dec 2009.

Examples

			The a(0)=1 null graph is vacuously 8-regular and connected; since it is acyclic then it has infinite girth.
The a(9)=1 graph is the complete bipartite graph K_{9,9} with 18 vertices.
The a(10)=1 graph has girth 4, automorphism group of order 7257600, and the following adjacency lists:
01 : 02 03 04 05 06 07 08 09 10
02 : 01 11 12 13 14 15 16 17 18
03 : 01 11 12 13 14 15 16 17 19
04 : 01 11 12 13 14 15 16 18 19
05 : 01 11 12 13 14 15 17 18 19
06 : 01 11 12 13 14 16 17 18 19
07 : 01 11 12 13 15 16 17 18 19
08 : 01 11 12 14 15 16 17 18 19
09 : 01 11 13 14 15 16 17 18 19
10 : 01 12 13 14 15 16 17 18 19
11 : 02 03 04 05 06 07 08 09 20
12 : 02 03 04 05 06 07 08 10 20
13 : 02 03 04 05 06 07 09 10 20
14 : 02 03 04 05 06 08 09 10 20
15 : 02 03 04 05 07 08 09 10 20
16 : 02 03 04 06 07 08 09 10 20
17 : 02 03 05 06 07 08 09 10 20
18 : 02 04 05 06 07 08 09 10 20
19 : 03 04 05 06 07 08 09 10 20
20 : 11 12 13 14 15 16 17 18 19

Links

Crossrefs

9-regular simple graphs with girth at least 4: this sequence (connected), A185294 (disconnected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), this sequence (k=9).
Connected 9-regular simple graphs with girth at least g: A014378 (g=3), this sequence (g=4).
Connected 9-regular simple graphs with girth exactly g: A184993 (g=3).

A184953 Number of connected 5-regular (or quintic) simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 59, 7847, 3459376, 2585136287, 2807104844073
Offset: 0

Views

Author

Jason Kimberley, Feb 27 2011

Keywords

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 3: A006923 (k=3), A184943 (k=4), this sequence (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: this sequence (g=3), A184954 (g=4), A184955 (g=5).

Formula

a(n) = A006821(n) - A058275(n).

A186733 Triangular array C(n,r) = number of connected r-regular graphs, having girth exactly 3, with n nodes, for 0 <= r < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 5, 3, 1, 1, 0, 0, 0, 0, 16, 0, 4, 0, 1, 0, 0, 0, 13, 57, 59, 21, 5, 1, 1, 0, 0, 0, 0, 263, 0, 266, 0, 6, 0, 1, 0, 0, 0, 63, 1532, 7847, 7848, 1547, 94, 9, 1, 1, 0, 0, 0, 0, 10747, 0, 367860, 0, 10786
Offset: 1

Views

Author

Jason Kimberley, Mar 26 2012

Keywords

Examples

			01: 0 ;
02: 0, 0 ;
03: 0, 0, 1 ;
04: 0, 0, 0, 1 ;
05: 0, 0, 0, 0, 1 ;
06: 0, 0, 0, 1, 1, 1 ;
07: 0, 0, 0, 0, 2, 0, 1 ;
08: 0, 0, 0, 3, 5, 3, 1, 1 ;
09: 0, 0, 0, 0, 16, 0, 4, 0, 1 ;
10: 0, 0, 0, 13, 57, 59, 21, 5, 1, 1 ;
11: 0, 0, 0, 0, 263, 0, 266, 0, 6, 0, 1 ;
12: 0, 0, 0, 63, 1532, 7847, 7848, 1547, 94, 9, 1, 1 ;
13: 0, 0, 0, 0, 10747, 0, 367860, 0, 10786, 0, 10, 0, 1 ;
14: 0, 0, 0, 399, 87948, 3459376, 21609299, 21609300, 3459386, 88193, 540, 13, 1, 1 ;
15: 0, 0, 0, 0, 803885, 0, 1470293674, 0, 1470293676, 0, 805579, 0, 17, 0, 1 ;
16: 0, 0, 0, 3268, 8020590, 2585136287, 113314233799, 733351105933, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Links

Crossrefs

The sum of the n-th row is A186743(n).
Connected k-regular simple graphs with girth exactly 3: this sequence (triangle), A186743 (any k); chosen k: A006923 (k=3), A184943 (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: A068934 (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: this sequence (g=3), A186734 (g=4).

Formula

C(n,r) = A068934(n,r) - A186714(n,r), noting that A186714 has 0 <= r <= n div 2.

A186743 Number of connected regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 13, 21, 157, 536, 18942, 389404, 50314456, 2942196832, 1698517018391
Offset: 0

Views

Author

Jason Kimberley, Dec 01 2011

Keywords

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 3: this sequence (any k), A186733 (triangular array); specified k: A006923 (k=3),A184943 (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7),A184983 (k=8), A184993 (k=9).

Formula

a(n) = A005177(n) - A186724(n).
Showing 1-6 of 6 results.

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

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