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 1-7 of 7 results.

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

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Author

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).

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

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

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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}.
		

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

A184964 Number of connected 6-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 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, Feb 28 2011

Keywords

Comments

Other than at n=0, this sequence first differs from A058276 at n = A054760(6,5) = 40.

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(12)=1 graph is the complete bipartite graph K_{6,6}.
		

Crossrefs

Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), A184944 (k=4), A184954 (k=5), this sequence (k=6), A184974 (k=7).
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: A184963 (g=3), this sequence (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

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).

A184973 Number of connected 7-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 1547, 21609300, 733351105933
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(4)=1 complete graph on 8 vertices is 7-regular; it has 28 edges and 56 triangles.
		

Crossrefs

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: this sequence (g=3), A184974 (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]]];
    A014377 = A@014377;
    A181153 = A@181153;
    a[n_] := A014377[[n + 1]] - A181153[[n + 1]];
    a /@ Range[0, 8] (* Jean-François Alcover, Jan 27 2020 *)

Formula

a(n) = A014377(n) - A181153(n).

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;
		

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;
		

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).
Showing 1-7 of 7 results.