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 11 results. Next

A165652 Number of disconnected 2-regular graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 8, 9, 12, 16, 20, 24, 32, 38, 48, 59, 72, 87, 109, 129, 157, 190, 229, 272, 330, 390, 467, 555, 659, 778, 926, 1086, 1283, 1509, 1774, 2074, 2437, 2841, 3322, 3871, 4509, 5236, 6094, 7055, 8181, 9464, 10944, 12624, 14577, 16778, 19322, 22209
Offset: 0

Views

Author

Jason Kimberley, Sep 28 2009

Keywords

Comments

a(n) is also the number of partitions of n such that each part i satisfies 2
For n>=2, it appears that a(n+1) is the number of (1,0)-separable partitions of n, as defined at A239482. For example, the four (1,0)-separable partitions of 9 are 621, 531, 441, 31212, corresponding to a(10) = 4. - Clark Kimberling, Mar 21 2014.

Examples

			The a(6)=1 graph is C_3+C_3. The a(7)=1 graph is C_3+C_4. The a(8)=2 graphs are C_3+C_5, C_4+C_4. The a(9)=3 graphs are 3C_3, C_3+C_6, C_4+C_5.

Links

Crossrefs

2-regular simple graphs: A179184 (connected), this sequence (disconnected), A008483 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), this sequence (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8).
Disconnected 2-regular simple graphs with girth at least g: this sequence (g=3), A185224 (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).
Cf. A239482.

Programs

  • Magma
    p := NumberOfPartitions; a := func< n | n lt 3 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3) - 1 >;

Formula

a = A008483 - A179184 = Euler_tranformation(A179184) - A179184.
For n > 2, since there is exactly one connected 2-regular graph on n vertices (the n cycle C_n) then a(n) = A008483(n) - 1.
(A008483(n) is also the number of not necessarily connected 2-regular graphs on n vertices.)
Column D(n, 2) in the triangle A068933.

A014378 Number of connected regular graphs of degree 8 with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848873, 214755319657939505395, 187549729101764460261498, 186685399408147545744203815, 210977245260028917322933154987
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Since the nontrivial 8-regular graph with the least number of vertices is K_9, there are no disconnected 8-regular graphs with less than 18 vertices. Thus for n<18 this sequence is identical to A180260. - Jason Kimberley, Sep 25 2009 and Feb 10 2011

Examples

			a(0)=1 because the null graph (with no vertices) is vacuously 8-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)
8-regular simple graphs: this sequence (connected), A165878 (disconnected), A180260 (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), A014377 (k=7), this sequence (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3). (End)

Formula

a(n) = A184983(n) + A181154(n).
a(n) = A180260(n) + A165878(n).
This sequence is the inverse Euler transformation of A180260.

Extensions

Using the symmetry of A051031, a(15) and a(16) were appended by Jason Kimberley, Sep 25 2009
a(17)-a(22) from Andrew Howroyd, Mar 13 2020

A033483 Number of disconnected 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 8, 25, 88, 378, 2026, 13351, 104595, 930586, 9124662, 96699987, 1095469608, 13175272208, 167460699184, 2241578965849, 31510542635443, 464047929509794, 7143991172244290, 114749135506381940, 1919658575933845129, 33393712487076999918, 603152722419661386031
Offset: 0

Views

Author

Ronald C. Read

Keywords

References

  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Links

Crossrefs

4-regular simple graphs: A006820 (connected), this sequence (disconnected), A033301 (not necessarily connected). - Jason Kimberley, Jan 08 2011
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), this sequence (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
Disconnected 4-regular simple graphs with girth at least g: this sequence (g=3), A185244 (g=4), A185245 (g=5), A185246 (g=6).

Programs

Formula

a(n) = A033301(n) - A006820(n) = Euler_transformation(A006820) - A006820.
a(n) = A068933(n, 4). - Jason Kimberley, Sep 27 2009 and Jan 08 2011

Extensions

Terms a(16)-a(18) from Martin Fuller, Dec 04 2006
Terms a(19)-a(26) from Jason Kimberley, Sep 27 2009 and Dec 30 2010
Terms a(27)-a(33), due to the extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

A165653 Number of disconnected 3-regular (cubic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 9, 31, 147, 809, 5855, 54477, 633057, 8724874, 137047391, 2391169355, 45626910415, 942659626031, 20937539944549, 497209670658529, 12566853576025106, 336749273734805530, 9534909974420181226
Offset: 0

Views

Author

Jason Kimberley, Sep 28 2009

Keywords

Links

Crossrefs

3-regular simple graphs: A002851 (connected), this sequence (disconnected), A005638 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), this sequence (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

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]]];
A005638 = A@005638;
A002851 = A@002851;
a[n_] := A005638[[n + 1]] - A002851[[n + 1]];
a /@ Range[0, 20] (* Jean-François Alcover, Jan 21 2020 *)

Formula

a(n) = A005638(n) - A002851(n).
a(n) = A068933(2n, 3).

A165655 Number of disconnected 5-regular (quintic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 66, 8029, 3484760, 2595985770, 2815099031417, 4230059694039460, 8529853839173455678, 22496718465713456081402, 75951258300080722467845995, 322269241532759484921710401976
Offset: 0

Views

Author

Jason Kimberley, Sep 28 2009

Keywords

Links

Crossrefs

5-regular simple graphs: A006821 (connected), this sequence (disconnected), A165626 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), this sequence (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

Formula

a = A165626 - A006821 = Euler_transformation(A006821) - A006821.
a(n)=A068933(2n,5).

Extensions

Terms a(13)-a(17), due to the extension of A006821 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

A165656 Number of disconnected 6-regular (sextic) graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 25, 297, 8199, 377004, 22014143, 1493574756, 114880777582, 9919463450855, 955388277929620, 102101882472479938, 12050526046888229845, 1563967741064673811531, 222318116370232302781485, 34486536277291555593662301, 5817920265098158804699762770
Offset: 0

Views

Author

Jason Kimberley, Sep 28 2009

Keywords

Links

Crossrefs

6-regular simple graphs: A006822 (connected), this sequence (disconnected), A165627 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), this sequence (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

Formula

a = A165627 - A006822 = Euler_transformation(A006822) - A006822.
a(n) = D(n, 6) in the triangle A068933.

Extensions

Terms a(25) and beyond from Andrew Howroyd, May 20 2020

A165877 Number of disconnected 7-regular (septic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 1562, 21617036, 733460349818, 42703733735064572, 4073409466378991404239, 613990076321940092226829047, 141518698937232822678583027258225
Offset: 0

Views

Author

Jason Kimberley, Sep 28 2009

Keywords

Links

Crossrefs

7-regular simple graphs: A014377 (connected), this sequence(disconnected), A165628 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), this sequence (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

Formula

a = A165628 - A014377 = Euler_transformation(A014377) - A014377.
a(n)=D(2n, 7) in the triangle A068933.

Extensions

a(13)-a(16) from Andrew Howroyd, May 20 2020

A185203 Number of disconnected 10-regular graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 11, 550, 806174, 2585947720, 9802278927562, 42709859521915286, 214798119408798346811, 1251607430636395979871600, 8463468717232507491862780325, 66406919318277846825588474735084
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

10-regular simple graphs: A014382 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), this sequence (k=10), A185213 (k=11).

Extensions

Terms a(29) and beyond from Andrew Howroyd, May 20 2020

A185213 Number of disconnected 11-regular graphs with 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 8037887, 945095928322681, 187549741420313256356540, 66398446859255608487987488813721, 43100445877221052008718432480116589483823
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

11-regular simple graphs: A014384 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), this sequence (k=11).

Extensions

a(15)-a(18) from Andrew Howroyd, May 20 2020

A185293 Number of disconnected 9-regular graphs with 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 88238, 113315027550, 281342192047999912, 1251394783006077652496450, 9854615127100313024544239975139, 134283364935428822131144679491097123786
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

9-regular simple graphs: A014381 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), this sequence (k=9), A185203 (k=10), A185213 (k=11).

Extensions

a(14)-a(17) from Andrew Howroyd, May 20 2020
Showing 1-10 of 11 results. Next

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

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