A185134 -id:A185134 - 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

A026797 Number of partitions of n in which the least part is 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889

Offset: 1

Clark Kimberling

a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - Jason Kimberley, Feb 22 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A008484.
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), A026796 (g=3), this sequence (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 4: A198314 (any k), A185644 (triangle); fixed k: this sequence (k=2), A185134 (k=3), A185144 (k=4).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^4/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^4/mul(1-x^(m+4), m=0..65), x, n+1), x, n), n = 1..60); # G. C. Greubel, Nov 03 2019

Table[Count[IntegerPartitions[n],?(Min[#]==4&)],{n,60}] (* _Harvey P. Dale, May 13 2012 *)
Rest@CoefficientList[Series[x^4/QPochhammer[x^4, x], {x,0,60}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^60)); concat([0,0,0], Vec(x^4/prod(m=0,70, 1-x^(m+4)))) \\ G. C. Greubel, Nov 03 2019

def A026797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list()
a=A026797_list(60); a[1:] # G. C. Greubel, Nov 03 2019

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

A185133 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 1, 1, 4, 15, 71, 428, 3406, 34270, 418621, 5937051, 94782437, 1670327647, 32090011476, 666351752261, 14859579573845

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 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), this sequence (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: this sequence (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A005638(n) - A185334(n).

a(n) = A006923(n) + A185033(n).

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.

A185144 Number of not necessarily 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, 16829, 193900, 2452820, 32670331, 456028487, 6636066126, 100135577863, 1582718910743

Offset: 0

Jason Kimberley, Nov 04 2011

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), A185134 (k=3), this sequence (k=4).
A185143 (g=3), A185144 (g=4).
Not necessarily connected 4-regular simple graphs with girth exactly g: A185140 (triangle); fixed g: A185143 (g=3), this sequence (g=4).

a(n) = A184944(n) + A185044(n) = A185140(n,4).

Corrected by Jason Kimberley, Jan 03 2013

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.

A198314 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 1, 8, 3, 37, 33, 335, 1610, 17985, 193911, 2867313, 32674066, 1581626531, 6705889862

Offset: 0

Jason Kimberley, Dec 12 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: this sequence (any k), A185644 (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

Not necessarily connected regular simple graphs girth exactly g: A198313 (g=3), this sequence (g=4), A198315 (g=5), A198316 (g=6), A198317 (g=7), A198318 (g=8).

a(n) = A186744(n) + A210714(n).

a(n) = A185314(n) - A185315(n).

a(10) corrected from 9 to 8 by Jason Kimberley, Feb 22 2013

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).

A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 21, 12, 1, 1, 0, 0, 2, 0, 31, 0, 0, 0, 0, 3, 103, 220, 7, 1, 1, 0, 0, 3, 0, 1606, 0, 1, 0, 0, 0, 5, 752, 16829, 388, 9, 1, 1, 0, 0, 5, 0, 193900, 0, 6, 0, 0, 0

Offset: 1

Jason Kimberley, Feb 22 2013

In the n-th row 0 <= 2k <= n.

			01: 0;
02: 0, 0;
03: 0, 0;
04: 0, 0, 1;
05: 0, 0, 0;
06: 0, 0, 0, 1;
07: 0, 0, 0, 0;
08: 0, 0, 1, 2, 1;
09: 0, 0, 1, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 1, 0, 2, 0;
12: 0, 0, 2, 21, 12, 1, 1;
13: 0, 0, 2, 0, 31, 0, 0;
14: 0, 0, 3, 103, 220, 7, 1, 1;
15: 0, 0, 3, 0, 1606, 0, 1, 0;
16: 0, 0, 5, 752, 16829, 388, 9, 1, 1;
17: 0, 0, 5, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 7, 7385, 2452820, 406824, 267, 8, 1, 1;
19: 0, 0, 8, 0, 32670331, 0, 3727, 0, 0, 0;
20: 0, 0, 11, 91939, 456028487, 1125022326, 483012, 741, 13, 1, 1;
21: 0, 0, 12, 0, 6636066126, 0, 69823723, 0, 1, 0, 0;
22: 0, 0, 16, 1345933, 100135577863, 3813549359275, 14836130862, 2887493, ?, 14, 1;

Jason Kimberley, Table of n, a(i)=E(n,k) for i = 1..139 (n = 1..22)
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 A198314(n).

Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), this sequence (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

E(n,k) = A186734(n,k) + A210704(n,k), noting the differing row lengths.

E(n,k) = A185304(n,k) - A185305(n,k), noting the differing row lengths.

cp's OEIS Frontend

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

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A026797 Number of partitions of n in which the least part is 4.

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A185133 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 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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A185144 Number of not necessarily connected 4-regular simple graphs on n vertices with girth exactly 4.

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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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A198314 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 4.

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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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