A186744 -id:A186744 - cp's OEIS frontend

A186734 Triangular array C(n,k) counting connected k-regular simple graphs on n vertices with girth exactly 4.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 20, 12, 1, 1, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 101, 220, 7, 1, 1, 0, 0, 0, 0, 1606, 0, 1, 0, 0, 0, 0, 743, 16828, 388, 9, 1, 1, 0, 0, 0, 0, 193900, 0, 6, 0, 0, 0, 0, 0, 7350

Offset: 1

Jason Kimberley, Mar 20 2013

In the n-th row 0 <= 2k <= n.

			01: 0;
02: 0, 0;
03: 0, 0;
04: 0, 0, 1;
05: 0, 0, 0;
06: 0, 0, 0, 1;
07: 0, 0, 0, 0;
08: 0, 0, 0, 2, 1;
09: 0, 0, 0, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 0, 0, 2, 0;
12: 0, 0, 0, 20, 12, 1, 1;
13: 0, 0, 0, 0, 31, 0, 0;
14: 0, 0, 0, 101, 220, 7, 1, 1;
15: 0, 0, 0, 0, 1606, 0, 1, 0;
16: 0, 0, 0, 743, 16828, 388, 9, 1, 1;
17: 0, 0, 0, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 0, 7350, 2452818, 406824, 267, 8, 1, 1;
19: 0, 0, 0, 0, 32670329, 0, 3727, 0, 0, 0;
20: 0, 0, 0, 91763, 456028472, 1125022325, 483012, 741, 13, 1, 1;
21: 0, 0, 0, 0, 6636066091, 0, 69823723, 0, 1, 0, 0;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A186744(n).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: A186733 (g=3), this sequence (g=4).
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).

C(n,k) = A186714(n,k) - A186715(n,k), noting the differing row lengths.

E(n,k) = A185644(n,k) - A210704(n,k), noting the differing row lengths.

A198314 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 1, 8, 3, 37, 33, 335, 1610, 17985, 193911, 2867313, 32674066, 1581626531, 6705889862

Offset: 0

Jason Kimberley, Dec 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 4: this sequence (any k), A185644 (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

Not necessarily connected regular simple graphs girth exactly g: A198313 (g=3), this sequence (g=4), A198315 (g=5), A198316 (g=6), A198317 (g=7), A198318 (g=8).

a(n) = A186744(n) + A210714(n).

a(n) = A185314(n) - A185315(n).

a(10) corrected from 9 to 8 by Jason Kimberley, Feb 22 2013

cp's OEIS Frontend

A186734 Triangular array C(n,k) counting connected k-regular simple graphs on n vertices with girth exactly 4.

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