A033301 -id:A033301 - cp's OEIS frontend

A008483 Number of partitions of n into parts >= 3.

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

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011
Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011
By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.
For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005
(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009
Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009
Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014
For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482.  For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014
From Peter Bala, Dec 01 2024: (Start)
Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)
Peter Adams, Saad I. El-Zanati, Peter Florido, and William Turner, On 2-Factorizations of the Complete 3-Uniform Hypergraph of Order 12 Minus a 1-Factor, Combinatorics, Graph Theory and Computing (SEICCGTC 2021) Springer Proc. Math. Stat., Vol 448, pp. 383-392. See p. 326.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
R.-Q. Feng, J. H. Kwak and E. K. Lloyd, Isomorphism classes of authentication codes, Bull. Austral. Math. Soc. 69 (2004), no. 2, 203-215.
Elisabeth Gaar and Daniel Krenn, Metamour-regular Polyamorous Relationships and Graphs, arXiv:2005.14121 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446
F. Jouneau-Sion and O. Torres, In Fisher's net: exact F-tests in semi-parametric models with exchangeable errors, August 2014, preprint on ResearchGate.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g.
Johan Kok, Degree affinity number of certain 2-regular graphs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
Eric Weisstein's World of Mathematics, Two-Regular Graph.

Essentially the same sequence as A026796 and A281356.
From Jason Kimberley, Nov 07 2009 and Jan 05 2011 and Feb 03 2011: (Start)
Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), this sequence (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected).
2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]).
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), this sequence (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), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10), ... (End)
Cf. A008284.

p := NumberOfPartitions; A008483 :=  func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; // Jason Kimberley, Jan 11 2011

series(1/product((1-x^i),i=3..50),x,51);
ZL := [ B,{B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); # Zerinvary Lajos, Mar 13 2007
with(combstruct):ZL2:=[S,{S=Set(Cycle(Z,card>2))}, unlabeled]:seq(count(ZL2,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
with(combstruct):a:=proc(m) [A,{A=Set(Cycle(Z,card>m))},unlabeled]; end: A008483:=a(2):seq(count(A008483,size=n),n=0..46); # Zerinvary Lajos, Oct 02 2007

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *)
Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]]  (* Clark Kimberling, Feb 27 2014 *)

a(n) = numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import partition
def A008483(n): return partition(n)-partition(n-1)-partition(n-2)+partition(n-3) # Chai Wah Wu, Jun 10 2025

a(n) = p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts (A000041).
G.f.: Product_{m>=3} 1/(1-x^m).
G.f.: (Sum_{n>=0} x^(3*n)) / (Product_{k=1..n} (1 - x^k)). - Joerg Arndt, Apr 17 2011
a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller, Aug 14 2006
Euler transformation of A179184. a(n) = A179184(n) + A165652(n). - Jason Kimberley, Jan 05 2011
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015
G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
a(n) = Sum_{j=0..floor(n/2)} A008284(n-2*j,j). - Gregory L. Simay, Apr 27 2023
G.f.: 1 + Sum_{n >= 1} x^(n+2)/Product_{k = 0..n-1} (1 - x^(k+3)). - Peter Bala, Dec 01 2024

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

N. J. A. Sloane

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

E. Friedman, Illustration of small graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Jennifer M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
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, Regular Graph.

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.

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002

Row sums of triangle A051031.

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

A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, 265, 1544, 10778, 88168, 805491, 8037418, 86221634, 985870522, 11946487647, 152808063181, 2056692014474, 29051272833609, 429668180677439, 6640165204855036, 107026584471569605, 1796101588825595008, 31333997930603283531, 567437240683788292989

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 2011
The Multiset Transform of this sequence gives a triangle which gives in row n and column k the 4-regular simple graphs with n>=1 nodes and k>=1 components (row sums A033301), starting:
;
;
;
;
1 ;
1 ;
2 ;
6 ;
16 ;
59 1 ;
265 1 ;
1544 3 ;
10778 8 ;
88168 25 ;
805491 87 1 ;
8037418 377 1 ;
86221634 2023 3 ;
985870522 13342 9 ;
11946487647 104568 27 ;
152808063181 930489 96 1 ;  - R. J. Mathar, Jun 02 2022

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.
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).

Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues, arXiv:2305.10562 [math.CO], 2023. See p. 7.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
M. Meringer, GenReg, Generation of regular graphs, program.
Markus Meringer, H. James Cleaves, Stephen J. Freeland, Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Quartic Graph
Eric Weisstein's World of Mathematics, Regular Graph
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

From Jason Kimberley, Mar 27 2010 and Jan 29 2011: (Start)
4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).
Connected regular simple graphs: A005177 (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), this sequence (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).
Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417  (multigraphs with loops verboten). (End)

a(n) = A184943(n) + A033886(n).
a(n) = A033301(n) - A033483(n).
Inverse Euler transform of A033301.
Row sums of A184940. - R. J. Mathar, May 30 2022

a(19)-a(22) were appended by Jason Kimberley on Sep 04 2009, Nov 24 2009, Mar 27 2010, and Mar 18 2011, from running M. Meringer's GENREG for 3.4, 44, and 403 processor days, and 15.5 processor years, at U. Ncle.

a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 10 2020

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

Original entry on oeis.org

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

A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 0, 4, 0, 16, 0, 4, 0, 1, 1, 1, 5, 21, 60, 60, 21, 5, 1, 1, 1, 0, 6, 0, 266, 0, 266, 0, 6, 0, 1, 1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1, 1, 0, 10, 0, 10786, 0, 367860, 0, 10786

Offset: 1

Eric W. Weisstein

A graph in which every node has r edges is called an r-regular graph. The triangle is symmetric because if an n-node graph is r-regular, than its complement is (n - 1 - r)-regular and two graphs are isomorphic if and only if their complements are isomorphic.

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A295193. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 08 2020

			T(8,3) = 6. Edge-lists for the 6 3-regular 8-node graphs:
  Graph 1: 12, 13, 14, 23, 24, 34, 56, 57, 58, 67, 68, 78
  Graph 2: 12, 13, 14, 24, 34, 26, 37, 56, 57, 58, 68, 78
  Graph 3: 12, 13, 23, 14, 47, 25, 58, 36, 45, 67, 68, 78
  Graph 4: 12, 13, 23, 14, 25, 36, 47, 48, 57, 58, 67, 68
  Graph 5: 12, 13, 24, 34, 15, 26, 37, 48, 56, 57, 68, 78
  Graph 6: 12, 23, 34, 45, 56, 67, 78, 18, 15, 26, 37, 48.
Triangle starts
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1,  1;
  1, 0, 1,  0,    1;
  1, 1, 2,  2,    1,    1;
  1, 0, 2,  0,    2,    0,    1;
  1, 1, 3,  6,    6,    3,    1,    1;
  1, 0, 4,  0,   16,    0,    4,    0,  1;
  1, 1, 5, 21,   60,   60,   21,    5,  1, 1;
  1, 0, 6,  0,  266,    0,  266,    0,  6, 0, 1;
  1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24, first 16 rows from Jason Kimberley)
Jason Kimberley, Not necessarily connected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Markus Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
Markus Meringer, Fast Generation of Regular Graphs and Construction of Cages, Univ. Bayreuth, 1999
Markus Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Regular Graph.

Row sums give A005176.
Regular graphs of degree k: A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Cf. A059441, A068933, A068934, A295193.

T(n,r) = A068934(n,r) + A068933(n,r).

More terms and comments from David Wasserman, Feb 22 2002
More terms from Eric W. Weisstein, Oct 19 2002
Description corrected (changed 'orders' to 'degrees') by Jason Kimberley, Sep 06 2009
Extended to the sixteenth row (in the b-file) by Jason Kimberley, Sep 24 2009

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

Ronald C. Read

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

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Quartic Graph

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).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] - A006820 (* Jean-François Alcover, Jan 21 2020, updated Mar 17 2020 *)

a(n) = A033301(n) - A006820(n) = Euler_transformation(A006820) - A006820.

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

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

A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 60, 7849, 3459386, 2585136741, 2807105258926, 4221456120848125, 8516994772686533749, 22470883220896245217626, 75883288448434648617038134, 322040154712674550886226182668

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-6)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms

5-regular simple graphs: A006821 (connected), A165655 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), specified degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), this sequence (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).

A006821 = Cases[Import["https://oeis.org/A006821/b006821.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006821] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transform of A006821.

Regular graphs cross-references edited by Jason Kimberley, Nov 07 2009
a(9) from Jason Kimberley, Nov 24 2009
a(10)-a(14) from Andrew Howroyd, Mar 10 2020

A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609301, 1470293676, 113314233813, 9799685588961, 945095823831333, 101114579937196179, 11945375659140003692, 1551593789610531820695, 220716215902794066709555, 34259321384370735003091907, 5782740798229835127025560294

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (n-7)-regular graphs on n vertices.

Georg Grasegger, Hakan Guler, Bill Jackson, Anthony Nixon, Flexible circuits in the d-dimensional rigidity matroid, arXiv:2003.06648 [math.CO], 2020.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph

6-regular simple graphs: A006822 (connected), A165656 (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), A005638 (k=3), A033301 (k=4), A165626 (k=5), this sequence (k=6), A165628 (k=7), A180260 (k=8).

A006822 = Cases[Import["https://oeis.org/A006822/b006822.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006822] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A006822.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(17) from Jason Kimberley, Dec 30 2010
a(18)-a(24) from Andrew Howroyd, Mar 07 2020

A165628 Number of 7-regular graphs (septic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 1547, 21609301, 733351105935, 42700033549946255, 4073194598236125134140, 613969628444792223023625238, 141515621596238755267618266465449

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-8)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Septic Graph

7-regular simple graphs: A014377 (connected), A165877 (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), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), this sequence (k=7), A180260 (k=8).

A014377 = Cases[Import["https://oeis.org/A014377/b014377.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014377] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014377.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(9)-a(11) from Andrew Howroyd, Mar 09 2020
a(12) from Andrew Howroyd, May 19 2020

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

cp's OEIS Frontend

A008483 Number of partitions of n into parts >= 3.

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A005176 Number of regular graphs with n unlabeled nodes.

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A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

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

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A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.

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

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A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

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A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

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