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 1-10 of 12 results. Next

A006822 Number of connected regular graphs of degree 6 (or sextic graphs) with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609300, 1470293675, 113314233808, 9799685588936, 945095823831036, 101114579937187980, 11945375659139626688, 1551593789610509806552, 220716215902792573134799, 34259321384370620122314325, 5782740798229825207562109439
Offset: 0

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Author

N. J. A. Sloane

Keywords

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

Links

Crossrefs

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
6-regular simple graphs: this sequence (connected), A165656 (disconnected), A165627 (not necessarily connected).
Connected regular graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), this sequence (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 6-regular simple graphs with girth at least g: this sequence (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4). (End)

Formula

a(n) = A184963(n) + A058276(n).
a(n) = A165627(n) - A165656(n).
This sequence is the inverse Euler transformation of A165627.

Extensions

a(16) and a(17) appended, from running M. Meringer's GENREG at U. Newcastle for 41 processor days and 3.5 processor years, by Jason Kimberley, Sep 04 2009 and Nov 13 2009.
Terms a(18)-a(24), due to the extension of A165627 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

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

Views

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

A006924 Number of connected trivalent graphs with 2n nodes and girth exactly 4.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 20, 101, 743, 7350, 91763, 1344782, 22160335, 401278984, 7885687604, 166870266608, 3781101495300
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 647.
  • Gordon Royle, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 4: this sequence (k=3), A184944 (k=4), A184954 (k=5), A184964 (k=6), A184974 (k=7).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), this sequence (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular simple graphs with girth at least g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

Formula

a(n) = A014371(n) - A014372(n).

Extensions

Definition corrected to include "connected", and "girth at least 4" minus "girth at least 5" formula provided by Jason Kimberley, Dec 12 2009

A058276 Number of connected 6-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862
Offset: 0

Views

Author

N. J. A. Sloane, Dec 17 2000

Keywords

Comments

The null graph on 0 vertices is vacuously connected and 6-regular; since it is acyclic, it has infinite girth. - Jason Kimberley, Jan 30 2011
Other than at n=0, this sequence first differs from A184964 at n = A054760(6,5) = 40.

Links

Crossrefs

6-regular simple graphs with girth at least 4: this sequence (connected), A185264 (disconnected), A185364 (not necessarily connected).
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), this sequence (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).
Connected 6-regular simple graphs with girth at least g: A006822 (g=3), this sequence (g=4).
Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4).

Formula

a(n) = A014377(n) - A184963(n).

Extensions

Terms a(19), a(20), and a(21), were appended, from running Meringer's GENREG at U. Ncle. for 51 processor days, by Jason Kimberley on Dec 11 2009
a(22) was appended, from running Meringer's GENREG at U. Ncle. for 1620 processor days, by Jason Kimberley on Dec 10 2011

A184944 Number of connected 4-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 12, 31, 220, 1606, 16828, 193900, 2452818, 32670329, 456028472, 6636066091, 100135577616, 1582718909051
Offset: 0

Views

Author

Jason Kimberley, Jan 26 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(8)=1 graph is the complete bipartite graph K_{4,4}.

Links

Crossrefs

4-regular simple graphs with girth exactly 4: this sequence (connected), A185044 (disconnected), A185144 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), this sequence (k=4), A184954 (k=5), A184964 (k=6), A184974 (k=7).
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: A184943 (g=3), this sequence (g=4), A184945 (g=5).

Formula

a(n) = A033886(n) - A058343(n).

Extensions

a(23) was appended by the author once A033886(23) was known, Nov 03 2011

A181153 Number of connected 7-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

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

Views

Author

Jason Kimberley, last week of Jan 2011

Keywords

Comments

a(10) was computed by the author in 3 hours using GENREG on Dec 02 2009.
a(11) was computed by the author using GENREG over 45.7 processor days at U. Newcastle from Jan 25 to 27 2011.

Examples

			The a(0)=1 null graph is vacuously 7-regular and connected; since it is acyclic then it has infinite girth.
The a(7)=1 graph is the complete bipartite graph K_{7,7} on 14 vertices.
The a(8)=1 graph has girth 4, automorphism group of order 80640, and the following adjacency lists:
01 : 02 03 04 05 06 07 08
02 : 01 09 10 11 12 13 14
03 : 01 09 10 11 12 13 15
04 : 01 09 10 11 12 14 15
05 : 01 09 10 11 13 14 15
06 : 01 09 10 12 13 14 15
07 : 01 09 11 12 13 14 15
08 : 01 10 11 12 13 14 15
09 : 02 03 04 05 06 07 16
10 : 02 03 04 05 06 08 16
11 : 02 03 04 05 07 08 16
12 : 02 03 04 06 07 08 16
13 : 02 03 05 06 07 08 16
14 : 02 04 05 06 07 08 16
15 : 03 04 05 06 07 08 16
16 : 09 10 11 12 13 14 15

References

  • M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146.

Links

Crossrefs

7-regular simple graphs with girth at least 4: this sequence (connected), A185274 (disconnected), A185374 (not necessarily connected).
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), this sequence (k=7), A181154 (k=8), A181170 (k=9).
Connected 7-regular simple graphs with girth at least g: A014377 (g=3), this sequence (g=4).
Connected 7-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4).

A184954 Number of connected 5-regular simple graphs on 2n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 7, 388, 406824, 1125022325, 3813549359274
Offset: 0

Views

Author

Jason Kimberley, Feb 27 2011

Keywords

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), A184944 (k=4), this sequence (k=5), A184964 (k=6), A184974 (k=7).
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: A184953 (g=3), this sequence (g=4), A184955 (g=5).

A184963 Number of connected 6-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7848, 367860, 21609299, 1470293674, 113314233799, 9799685588930
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 6-regular and connected, since it is acyclic, it has infinite girth.
The a(7)=1 complete graph on 7 vertices is 6-regular; it has 21 edges and 35 triangles.

Links

Crossrefs

Connected 6-regular simple graphs with girth at least g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: this sequence (g=3), A184964 (g=4).

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]]];
A006822 = A@006822;
A058276 = A@058276;
a[n_] := A006822[[n + 1]] - A058276[[n + 1]];
a /@ Range[0, 17] (* Jean-François Alcover, Jan 27 2020 *)

Formula

a(n) = A006822(n) - A058276(n).

A185364 Not necessarily connected 6-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862
Offset: 0

Views

Author

Jason Kimberley, Dec 07 2011

Keywords

Comments

First differs from A058276 at n=24.

Links

Crossrefs

6-regular simple graphs with girth at least 4: A058276 (connected), A185264 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), this sequence (k=6).
Cf. A184964.

Programs

Formula

This sequence is the Euler transformation of A058276.
a(n) = A058276(n) + A185264(n).

A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 4, 21, 266, 7848, 1, 367860, 0, 21609299, 1, 1470293674, 1, 113314233799, 9, 9799685588930, 6
Offset: 7

Views

Author

Jason Kimberley, Feb 24 2011

Keywords

Comments

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).

Examples

			Triangle begins:
1;
1;
4;
21;
266;
7848, 1;
367860, 0;
21609299, 1;
1470293674, 1;
113314233799, 9;
9799685588930, 6;
?, 267;
?, 3727;
?, 483012;
?, 69823723;
?, 14836130862;
The C(40,5)=1 (see the a-file) graph, the unique (6,5)-cage, is the complement of a Petersen graph inside the Hoffman-Singleton graph [from Brouwer link].
The first known of C(42,5)>=1 graph(s) has automorphism group of order 5040 and these adjacency lists:
1 : 2 3 4 5 6 7
2 : 1 8 9 10 11 12
3 : 1 13 14 15 16 17
4 : 1 18 19 20 21 22
5 : 1 23 24 25 26 27
6 : 1 28 29 30 31 32
7 : 1 33 34 35 36 37
8 : 2 13 18 23 28 38
9 : 2 14 19 24 33 39
10 : 2 15 20 29 34 40
11 : 2 16 25 30 35 41
12 : 2 21 26 31 36 42
13 : 3 8 21 27 34 41
14 : 3 9 26 28 37 40
15 : 3 10 22 25 31 39
16 : 3 11 19 32 36 38
17 : 3 20 23 30 33 42
18 : 4 8 25 32 33 40
19 : 4 9 16 27 29 42
20 : 4 10 17 26 35 38
21 : 4 12 13 30 37 39
22 : 4 15 24 28 36 41
23 : 5 8 17 29 36 39
24 : 5 9 22 30 34 38
25 : 5 11 15 18 37 42
26 : 5 12 14 20 32 41
27 : 5 13 19 31 35 40
28 : 6 8 14 22 35 42
29 : 6 10 19 23 37 41
30 : 6 11 17 21 24 40
31 : 6 12 15 27 33 38
32 : 6 16 18 26 34 39
33 : 7 9 17 18 31 41
34 : 7 10 13 24 32 42
35 : 7 11 20 27 28 39
36 : 7 12 16 22 23 40
37 : 7 14 21 25 29 38
38 : 8 16 20 24 31 37
39 : 9 15 21 23 32 35
40 : 10 14 18 27 30 36
41 : 11 13 22 26 29 33
42 : 12 17 19 25 28 34

Links

Crossrefs

Connected 6-regular simple graphs with girth at least g: A184961 (triangle); chosen g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184963 (g=3), A184964 (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), this sequence (k=6), A184970 (k=7), A184980 (k=8).

Extensions

After approximately 390 processor days of computation with genreg, C(41,5)=0.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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