A185133 -id:A185133 - cp's OEIS frontend

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A026796 Number of partitions of n in which the least part is 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510, 5237, 6095, 7056, 8182, 9465, 10945, 12625

Offset: 0

Clark Kimberling

Let b(k) be the number of partitions of k for which twice the number of ones is the number of parts, k = 0, 1, 2, ... . Then a(n+4) = b(n), n = 0, 1, 2, ... (conjectured). - George Beck, Aug 19 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same sequence as A008483.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), this sequence (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: this sequence (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^3/(&*[1-x^(m+3): m in [0..70]]) )); // G. C. Greubel, Nov 02 2019

seq(coeff(series(x^3/mul(1-x^(m+3), m=0..65), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Nov 02 2019

Table[Count[IntegerPartitions[n], p_ /; Min@p==3], {n, 0, 60}] (* George Beck Aug 19 2017 *)
CoefficientList[Series[x^3/QPochhammer[x^3, x], {x,0,60}], x] (* G. C. Greubel, Nov 02 2019 *)

a(n) = numbpart(n-3) - numbpart(n-4) - numbpart(n-5) + numbpart(n-6); \\ Michel Marcus, Aug 20 2014

x='x+O('x^66); Vecrev(Pol(x^3*(1-x)*(1-x^2)/eta(x))) \\ Joerg Arndt, Aug 22 2014

def A026796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^3/product((1-x^(m+3)) for m in (0..65)) ).list()
A026796_list(60) # G. C. Greubel, Nov 02 2019

G.f.: x^3 / Product_{m>=3} (1 - x^m).
a(n) = p(n-3) - p(n-4) - p(n-5) + p(n-6), where p(n) = A000041(n). - Bob Selcoe, Aug 07 2014
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(3*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Michel Marcus, Aug 20 2014

a(0) = 0 prepended by Joerg Arndt, Aug 22 2014

A185334 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 23, 112, 801, 7840, 97723, 1436873, 23791155, 432878091, 8544173926, 181519645163, 4127569521160

Offset: 0

Jason Kimberley, Feb 15 2011

The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 4: A014371 (connected), A185234 (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), this sequence (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), this sequence (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

A014371 = Cases[Import["https://oeis.org/A014371/b014371.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014371] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014371.

A185134 Number of, not necessarily connected, 3-regular simple graphs on 2n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 21, 103, 752, 7385, 91939, 1345933, 22170664, 401399440, 7887389438, 166897766824, 3781593764772

Offset: 0

Jason Kimberley, Mar 21 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: A026797 (k=2), this sequence (k=3), A185144 (k=4).

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), this sequence (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A185334(n) - A185335(n).

a(n) = A006924(n) + A185034(n).

A198313 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 15, 23, 162, 540, 18958, 389417, 50314520, 2942196930, 1698517018988

Offset: 0

Jason Kimberley, May 24 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 3: this sequence (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

Not necessarily connected regular simple graphs girth exactly g: this sequence (g=3), A198314 (g=4), A198315 (g=5), A198316 (g=6), A198317 (g=7), A198318 (g=8).

a(n) = A186743(n) + A210713(n).
a(n) = A005176(n) - A185314(n).
a(n) is the sum of the n-th row of A185643.

A185143 Number of not necessarily 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, 58, 264, 1535, 10755, 87973, 803973, 8020967, 86029760, 983431053, 11913921910, 152352965278, 2050065073002, 28951233955602, 428086557232387

Offset: 0

Jason Kimberley, Mar 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

4-regular simple graphs with girth exactly 3: A184943 (connected), A185043 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), this sequence (k=4), A185153 (k=5), A185163 (k=6).
Not necessarily connected 4-regular simple graphs with girth exactly g: A185140 (triangle); fixed g: this sequence (g=3), A185144 (g=4).

a(n) = A033301(n) - A185344(n).

a(n) = A184943(n) + A185043(n).

a(22) corrected and a(23) appended, due to the correction and extension of A033301 by Andrew Howroyd, from Jason Kimberley, Mar 14 2020

A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 9, 49, 455, 5784, 90940, 1620491, 31478651, 656784488, 14621878339, 345975756388

Offset: 0

Jason Kimberley, Jan 28 2011

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 5: A014372 (connected), A185235 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), this sequence (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: A185325 (k=2), this sequence (k=3).

A014372 = Cases[Import["https://oeis.org/A014372/b014372.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014372] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A014372.

A185135 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 8, 48, 450, 5752, 90555, 1612917, 31297424, 652159986, 14499787794, 342646826428

Offset: 0

Jason Kimberley, Mar 21 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), A185134 (g=4), this sequence (g=5), A185136 (g=6).

a(n) = A185335(n) - A185336(n).

a(n) = A006925(n) + A185035(n).

A185136 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7573, 181224, 4624481, 122089999, 3328899592, 93988909792

Offset: 0

Jason Kimberley, Mar 21 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), A185134 (g=4), A185135 (g=5), this sequence (g=6).

a(n) = A006926(n) + A185036(n).

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 4, 5, 3, 1, 1, 0, 0, 2, 0, 16, 0, 4, 0, 1, 0, 0, 2, 15, 58, 59, 21, 5, 1, 1, 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1, 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1, 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1

Offset: 1

Jason Kimberley, Feb 07 2013

			01: 0;
02: 0, 0;
03: 0, 0, 1;
04: 0, 0, 0, 1;
05: 0, 0, 0, 0, 1;
06: 0, 0, 1, 1, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 4, 5, 3, 1, 1;
09: 0, 0, 2, 0, 16, 0, 4, 0, 1;
10: 0, 0, 2, 15, 58, 59, 21, 5, 1, 1;
11: 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1;
12: 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1;
13: 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 6, 428, 87973, 3459379, 21609300, 21609300, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 9, 0, 803973, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 10, 3406, 8020967, 2585136353, 113314233804, 733351105934, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Jason Kimberley, Table of i, a(i)=E(n,k) for i = 1..136 (n = 1..16)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A198313(n).

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), this sequence (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

E(n,k) = A186733(n,k) + A210703(n,k), noting that A210703 is a tabf.

E(n,k) = A051031(n,k) - A185304(n,k), noting that A185304 is a tabf.

cp's OEIS Frontend

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

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A026796 Number of partitions of n in which the least part is 3.

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A185334 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.

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A185134 Number of, not necessarily connected, 3-regular simple graphs on 2n vertices with girth exactly 4.

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A198313 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 3.

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A185143 Number of not necessarily connected 4-regular simple graphs on n vertices with girth exactly 3.

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A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

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A185135 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 5.

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A185136 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 6.

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