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%I #12 May 01 2014 02:37:07
%S 0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,1,1,1,0,0,0,0,2,0,1,0,0,0,3,5,3,
%T 1,1,0,0,0,0,16,0,4,0,1,0,0,0,13,57,59,21,5,1,1,0,0,0,0,263,0,266,0,6,
%U 0,1,0,0,0,63,1532,7847,7848,1547,94,9,1,1,0,0,0,0,10747,0,367860,0,10786
%N Triangular array C(n,r) = number of connected r-regular graphs, having girth exactly 3, with n nodes, for 0 <= r < n.
%H Jason Kimberley, <a href="/A186733/b186733.txt">Table of i, a(i)=C(n,r) for i = 1..136 (n = 1..16)</a>
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_eq_g_index">Index of sequences counting connected k-regular simple graphs with girth exactly g</a>
%F C(n,r) = A068934(n,r) - A186714(n,r), noting that A186714 has 0 <= r <= n div 2.
%e 01: 0 ;
%e 02: 0, 0 ;
%e 03: 0, 0, 1 ;
%e 04: 0, 0, 0, 1 ;
%e 05: 0, 0, 0, 0, 1 ;
%e 06: 0, 0, 0, 1, 1, 1 ;
%e 07: 0, 0, 0, 0, 2, 0, 1 ;
%e 08: 0, 0, 0, 3, 5, 3, 1, 1 ;
%e 09: 0, 0, 0, 0, 16, 0, 4, 0, 1 ;
%e 10: 0, 0, 0, 13, 57, 59, 21, 5, 1, 1 ;
%e 11: 0, 0, 0, 0, 263, 0, 266, 0, 6, 0, 1 ;
%e 12: 0, 0, 0, 63, 1532, 7847, 7848, 1547, 94, 9, 1, 1 ;
%e 13: 0, 0, 0, 0, 10747, 0, 367860, 0, 10786, 0, 10, 0, 1 ;
%e 14: 0, 0, 0, 399, 87948, 3459376, 21609299, 21609300, 3459386, 88193, 540, 13, 1, 1 ;
%e 15: 0, 0, 0, 0, 803885, 0, 1470293674, 0, 1470293676, 0, 805579, 0, 17, 0, 1 ;
%e 16: 0, 0, 0, 3268, 8020590, 2585136287, 113314233799, 733351105933, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;
%Y The sum of the n-th row is A186743(n).
%Y Connected k-regular simple graphs with girth exactly 3: this sequence (triangle), A186743 (any k); chosen k: A006923 (k=3), A184943 (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
%Y Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: A068934 (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
%Y Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: this sequence (g=3), A186734 (g=4).
%K nonn,tabl,hard
%O 1,26
%A _Jason Kimberley_, Mar 26 2012
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