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.
%I A051031 #63 Jul 08 2025 10:40:10 %S A051031 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, %T A051031 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, %U A051031 0,1,1,1,9,94,1547,7849,7849,1547,94,9,1,1,1,0,10,0,10786,0,367860,0,10786 %N A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n. %C A051031 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. %C A051031 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 %H A051031 Andrew Howroyd, <a href="/A051031/b051031.txt">Table of n, a(n) for n = 1..300</a> (rows 1..24, first 16 rows from Jason Kimberley) %H A051031 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/A051031">Not necessarily connected regular graphs (with girth at least 3)</a> %H A051031 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_ge_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g</a> %H A051031 Markus Meringer, <a href="http://dx.doi.org/10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G">Fast generation of regular graphs and construction of cages</a>, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009] %H A051031 Markus Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/pdf/pub/FastGenRegGraphJGT.pdf">Fast Generation of Regular Graphs and Construction of Cages</a>, Univ. Bayreuth, 1999 %H A051031 Markus Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a> %H A051031 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RegularGraph.html">Regular Graph.</a> %F A051031 T(n,r) = A068934(n,r) + A068933(n,r). %e A051031 T(8,3) = 6. Edge-lists for the 6 3-regular 8-node graphs: %e A051031 Graph 1: 12, 13, 14, 23, 24, 34, 56, 57, 58, 67, 68, 78 %e A051031 Graph 2: 12, 13, 14, 24, 34, 26, 37, 56, 57, 58, 68, 78 %e A051031 Graph 3: 12, 13, 23, 14, 47, 25, 58, 36, 45, 67, 68, 78 %e A051031 Graph 4: 12, 13, 23, 14, 25, 36, 47, 48, 57, 58, 67, 68 %e A051031 Graph 5: 12, 13, 24, 34, 15, 26, 37, 48, 56, 57, 68, 78 %e A051031 Graph 6: 12, 23, 34, 45, 56, 67, 78, 18, 15, 26, 37, 48. %e A051031 Triangle starts %e A051031 1; %e A051031 1, 1; %e A051031 1, 0, 1; %e A051031 1, 1, 1, 1; %e A051031 1, 0, 1, 0, 1; %e A051031 1, 1, 2, 2, 1, 1; %e A051031 1, 0, 2, 0, 2, 0, 1; %e A051031 1, 1, 3, 6, 6, 3, 1, 1; %e A051031 1, 0, 4, 0, 16, 0, 4, 0, 1; %e A051031 1, 1, 5, 21, 60, 60, 21, 5, 1, 1; %e A051031 1, 0, 6, 0, 266, 0, 266, 0, 6, 0, 1; %e A051031 1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1; %e A051031 ... %Y A051031 Row sums give A005176. %Y A051031 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). %Y A051031 Cf. A059441, A068933, A068934, A295193. %K A051031 nonn,tabl %O A051031 1,18 %A A051031 _Eric W. Weisstein_ %E A051031 More terms and comments from _David Wasserman_, Feb 22 2002 %E A051031 More terms from _Eric W. Weisstein_, Oct 19 2002 %E A051031 Description corrected (changed 'orders' to 'degrees') by _Jason Kimberley_, Sep 06 2009 %E A051031 Extended to the sixteenth row (in the b-file) by _Jason Kimberley_, Sep 24 2009