A165648 -id:A165648 - cp's OEIS frontend

A005177 Number of connected regular graphs with n nodes.

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

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
Daniel R. Herber, Enhancements to the perfect matching approach for graph enumeration-based engineering challenges, Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2020).
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Markus Meringer, GENREG: A program for Connected Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Sep 23 2009]
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.

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

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.

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

A165647 Number of simple graphs on n vertices with each component regular.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 27, 58, 99, 316, 936, 20225, 410571, 50745729, 2993355213, 1701561156737, 444491521018441, 650424404724148930, 430363737398835017296, 2886485242642120438501318, 8838475960755164592843156768, 152938120727373876718042362054465

Offset: 1

Jason Kimberley, Sep 23 2009

			The a(1)=1 graph is: K_1.
The a(2)=2 graphs are: 2K_1, K_2.
The a(3)=3 graphs are: 3K_1, K_1+K_2, K_3.
The a(4)=6 graphs are: 4K_1, 2K_1+K_2, K_1+K_3, 2K_2, C_4, K_4.

N. J. A. Sloane, Transforms

Row sums of A275420.

A005177 = Cases[Import["https://oeis.org/A005177/b005177.txt", "Table"], {, }][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[DivisorSum[j, # p[#]&] b[n - j], {j, 1, n}]/n]; b];
a = etr[A005177[[#+1]]&];
a /@ Range[16] (* Jean-François Alcover, Dec 02 2019 *)

This sequence is the Euler transformation of A005177.

a(n) = A005177(n) + A165648(n).

Terms a(17) and beyond from Andrew Howroyd, May 20 2020

cp's OEIS Frontend

A005177 Number of connected regular graphs with n nodes.

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