A185204 -id:A185204 - cp's OEIS frontend

A210703 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth exactly 3.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 0, 3, 0, 1, 0, 0, 4, 8, 3, 1, 0, 0, 5, 0, 8, 0, 0, 0, 6, 29, 25, 3, 1, 0, 0, 9, 0, 88, 0, 1, 0, 0, 10, 138, 377, 66, 5, 1, 0, 0, 13, 0, 2026, 0, 25, 0, 0, 0, 17, 774, 13349, 8029, 297, 5, 1, 0, 0, 21, 0, 104593, 0, 8199, 0, 1, 0, 0, 25, 5678, 930571, 3484759, 377004, 1562, 7, 1

Offset: 2

Jason Kimberley, Jan 21 2013

			2: 0;
3: 0;
4: 0, 0;
5: 0, 0;
6: 0, 0, 1;
7: 0, 0, 1;
8: 0, 0, 1, 1;
9: 0, 0, 2, 0;
10: 0, 0, 2, 2, 1;
11: 0, 0, 3, 0, 1;
12: 0, 0, 4, 8, 3, 1;
13: 0, 0, 5, 0, 8, 0;
14: 0, 0, 6, 29, 25, 3, 1;
15: 0, 0, 9, 0, 88, 0, 1;
16: 0, 0, 10, 138, 377, 66, 5, 1;
17: 0, 0, 13, 0, 2026, 0, 25, 0;
18: 0, 0, 17, 774, 13349, 8029, 297, 5, 1;
19: 0, 0, 21, 0, 104593, 0, 8199, 0, 1;
20: 0, 0, 25, 5678, 930571, 3484759, 377004, 1562, 7, 1;
21: 0, 0, 33, 0, 9124627, 0, 22014143, 0, 100, 0;
22: 0, 0, 39, 53324, 96699740, 2595985769, 1493574756, 21617036, 10901, 9, 1;
23: 0, 0, 49, 0, 1095467916, 0, 114880777582, 0, 3470736, 0, 1;
24: 0, 0, 60, 622716, 13175254799, 2815099031409, 9919463450854, 733460349818, 1473822243, 88238, 11, 1;

Jason Kimberley, Table of i, a(i)=D(n,k) for i = 2..145 (n = 2..24)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth exactly g

The sum of the n-th row is A210713(n).

Disconnected k-regular simple graphs with girth exactly 3: A210713 (any k), this sequence (triangle); for a fixed k: A185033 (k=3), A185043 (k=4), A185053 (k=5), A185063 (k=6).

D(n,k) = A068933(n,k) - A185204(n,k) [the former is padded to be a tabl but the latter is a tabf].

D(n,k) = A185643(n,k) - A186733(n,k) [both are tabl but the result is tabf].

A185200 Square array O(k,g) containing the least nontrivial order of disconnected k-regular simple graphs with girth at least g.

Original entry on oeis.org

6, 8, 8, 10, 12, 10, 12, 16, 20, 12, 14, 20, 38, 28, 14, 16, 24, 60, 52, 48, 16, 18, 28, 80, 84, 134, 60, 18, 20, 32, 100, 124

Offset: 0

Jason Kimberley, Jan 25 2011

This array is twice A054760.

(A proof of the monotonicity with respect to girth appears in Holton & Sheehan)

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Cf. A185204 - A185207: the triangular arrays D(n,k) counting disconnected k-regular simple graphs with girth at least g on n vertices, for 4 <= g <= 7.

cp's OEIS Frontend

A210703 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth exactly 3.

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