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 1-8 of 8 results.

A014377 Number of connected regular graphs of degree 7 with 2n nodes.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 1547, 21609301, 733351105934, 42700033549946250, 4073194598236125132578, 613969628444792223002008202, 141515621596238755266884806115631
Offset: 0

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Author

N. J. A. Sloane

Keywords

Examples

			a(0)=1 because the null graph (with no vertices) is vacuously 7-regular and connected.

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

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
7-regular simple graphs: this sequence (connected), A165877 (disconnected), A165628 (not necessarily connected).
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), this sequence (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 7-regular simple graphs with girth at least g: this sequence (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4), A184965 (g=5). (End)

Formula

a(n) = A184973(n) + A181153(n).
a(n) = A165628(n) - A165877(n).
This sequence is the inverse Euler transformation of A165628.

Extensions

Added another term from Meringer's page. Dmitry Kamenetsky, Jul 28 2009
Term a(8) (on Meringer's page) was found from running Meringer's GENREG for 325 processor days at U. Newcastle by Jason Kimberley, Oct 02 2009
a(9)-a(11) from Andrew Howroyd, Mar 13 2020
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

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.

A184970 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 5, 1547, 21609300, 1, 733351105933, 1
Offset: 4

Views

Author

Jason Kimberley, Feb 25 2011

Keywords

Comments

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

Examples

			1;
5;
1547;
21609300, 1;
733351105933, 1;
?, 8;
?, 741;
?, 2887493;

Links

Crossrefs

Connected 7-regular simple graphs with girth at least g: A184971 (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184973 (g=3), A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), this sequence (k=7), A184980 (k=8).

A184971 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 5, 1547, 21609301, 1, 733351105934, 1
Offset: 4

Views

Author

Jason Kimberley, Jan 10 2012

Keywords

Comments

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

Examples

			1;
5;
1547;
21609301, 1;
733351105934, 1;
?, 8;
?, 741;
?, 2887493;

Links

Crossrefs

Connected 7-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184970 (triangle); chosen g: A184973 (g=3), A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), A184941 (k=4), A184951 (k=5), A184961 (k=6), this sequence (k=7), A184981 (k=8).

A184974 Number of connected 7-regular simple graphs on 2n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 8, 741, 2887493
Offset: 0

Views

Author

Jason Kimberley, Feb 28 2011

Keywords

Examples

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 7-regular and connected, since it is acyclic, it has infinite girth.
The a(7)=1 graph is the complete bipartite graph K_{7,7}.

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), A184944 (k=4), A184954 (k=5), A184964 (k=6), this sequence (k=7).
Connected 7-regular simple graphs with girth at least g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184973 (g=3), this sequence (g=4).

Formula

a(n) = A186714(n,5) - A186715(n,5).

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 1-8 of 8 results.

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

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