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 19 results. Next

A005176 Number of regular graphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, 546, 19002, 389454, 50314870, 2942198546, 1698517037030, 442786966117636, 649978211591622812, 429712868499646587714, 2886054228478618215888598, 8835589045148342277802657274, 152929279364927228928025482936226, 1207932509391069805495173417972533120, 99162609848561525198669168653641835566774
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Simple regular graphs of any degree: A005177 (connected), A068932 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs with girth at least g: this sequence (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).
Cf. A295193.

Formula

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002
Row sums of triangle A051031.

Extensions

More terms from David Wasserman, Mar 08 2002
a(15) and a(16) from Jason Kimberley, Sep 25 2009
Edited by Jason Kimberley, Jan 06 2011 and May 24 2012
a(17)-a(21) from Andrew Howroyd, Mar 08 2020
a(22)-a(24) from Andrew Howroyd, Apr 05 2020

A005177 Number of connected regular graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, 539, 18979, 389436, 50314796, 2942198440, 1698517036411, 442786966115560, 649978211591600286, 429712868499646474880, 2886054228478618211088773, 8835589045148342277771518309, 152929279364927228928021274993215, 1207932509391069805495173301992815105, 99162609848561525198669168640159162918815
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Regular simple graphs of any degree: this sequence (connected), A068932 (disconnected), A005176 (not necessarily connected), A275420 (multisets).
Connected regular graphs of any degree with girth at least g: this sequence (g=3), A186724 (g=4), A186725 (g=5), A186726 (g=6), A186727 (g=7), A186728 (g=8), A186729 (g=9).
Connected regular simple graphs: this sequence (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), A014381 (k=9), A014382 (k=10), A014384 (k=11). - Jason Kimberley, Nov 03 2011

Formula

a(n) = sum of the n-th row of A068934.
a(n) = A165647(n) - A165648(n).
This sequence is the inverse Euler transformation of A165647.

Extensions

More terms from David Wasserman, Mar 08 2002
a(15) from Giovanni Resta, Feb 05 2009
Terms are sums of the output from M. Meringer's genreg software. To complete a(16) it was run by Jason Kimberley, Sep 23 2009
a(0)=1 (due to the empty graph being vacuously connected and regular) inserted by Jason Kimberley, Apr 11 2012
a(17)-a(21) from Andrew Howroyd, Mar 10 2020
a(22)-a(24) from Andrew Howroyd, May 19 2020

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

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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.

A068933 Triangular array D(n, r) = number of disconnected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 1, 4, 2, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 12, 31, 25, 3, 1, 0, 0, 0, 0, 0
Offset: 1

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Author

David Wasserman, Mar 08 2002

Keywords

Comments

A graph is called r-regular if every node has exactly r edges. Row sums give A068932.

Examples

			This sequence can be computed using the information in A068934. We'll abbreviate A068934(n, r) as C(n, r). To compute D(13, 4), note that the connected components of a 4-regular graph must have at least 5 elements. So a disconnected 13-node 4-regular graph must have two components and their sizes are either 8 and 5, or 7 and 6. So D(13, 4) = C(8, 4)*C(5, 4) + C(7, 4)*C(6, 4) = 6*1 + 2*1 = 8.
0;
1, 0;
1, 0, 0;
1, 1, 0, 0;
1, 0, 0, 0, 0;
1, 1, 1, 0, 0, 0;
1, 0, 1, 0, 0, 0, 0;
1, 1, 2, 1, 0, 0, 0, 0;
1, 0, 3, 0, 0, 0, 0, 0, 0;
1, 1, 4, 2, 1, 0, 0, 0, 0, 0;
1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0;
1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0;
1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0;
1, 1, 12, 31, 25, 3...

Links

Crossrefs

Cf. A051031, A068932, A068934.

Formula

D(n, r) = A051031(n, r) - A068934(n, r).

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

A165878 Number of disconnected 8-regular simple graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 100, 10901, 3470736, 1473822243, 734843169811, 423929978716908, 281768931380519766, 215039290728074333738, 187766225244288486398132, 186874272297562916477691894, 211165081721567703008217979077
Offset: 0

Views

Author

Jason Kimberley, Sep 29 2009

Keywords

Examples

			The a(18)=1 graph is K_9+K_9.

Links

Crossrefs

8-regular simple graphs: A014378 (connected), this sequence (disconnected), A180260 (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), A165877 (k=7), this sequence (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

Formula

a = A180260 - A014378 = Euler_transformation(A014378) - A014378.
a(n) = D(n, 8) in the triangle A068933.

Extensions

Terms a(26) and beyond from Andrew Howroyd, May 20 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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