A185133 -id:A185133 - cp's OEIS frontend

A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

0, 0, 0, 1, 3, 59, 7848, 3459379, 2585136353, 2807104852102

Offset: 0

Jason Kimberley, Mar 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 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), this sequence (k=5), A185163 (k=6).

a(n) = A165626(n) - A185354(n).

a(n) = A184953(n) + A185053(n).

A185163 Number of not necessarily connected 6-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7848, 367860, 21609300, 1470293675, 113314233804, 9799685588955

Offset: 0

Jason Kimberley, Mar 21 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 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), this sequence (k=6).

a(n) = A165627(n) - A185364(n).

a(n) = A184953(n) + A185053(n).

A185130 Irregular triangle E(n,g) counting not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 1, 4, 2, 15, 5, 1, 71, 21, 2, 428, 103, 8, 1, 3406, 752, 48, 1, 34270, 7385, 450, 5, 418621, 91939, 5752, 32, 5937051, 1345933, 90555, 385, 94782437, 22170664, 1612917, 7573, 1, 1670327647, 401399440, 31297424, 181224, 3, 32090011476, 7887389438

Offset: 2

Jason Kimberley, Dec 26 2012

The first column is for girth exactly 3. The column for girth exactly g begins when 2n reaches A000066(g).

			1;
1, 1;
4, 2;
15, 5, 1;
71, 21, 2;
428, 103, 8, 1;
3406, 752, 48, 1;
34270, 7385, 450, 5;
418621, 91939, 5752, 32;
5937051, 1345933, 90555, 385;
94782437, 22170664, 1612917, 7573, 1;
1670327647, 401399440, 31297424, 181224, 3;
32090011476, 7887389438, 652159986, 4624481, 21;
666351752261, 166897766824, 14499787794, 122089999, 545, 1;
14859579573845, 3781593764772, 342646826428, 3328899592, 30368, 0;

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

Initial columns of this triangle: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

The n-th row is the sequence of differences of the n-th row of A185330:
E(n,g) = A185330(n,g) - A185330(n,g+1), once we have appended 0 to each row of A185330.
Hence the sum of the n-th row is A185330(n,3) = A005638(n).

A185336 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7574, 181227, 4624502, 122090545, 3328929960, 93990692632, 2754222605808

Offset: 0

Jason Kimberley, Jan 28 2012

The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.

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

3-regular simple graphs with girth at least 6: A014374 (connected), A185236 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 6: A185326 (k=2), this sequence (k=3).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), A185335 (g=5), this sequence (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

A014374 = Cases[Import["https://oeis.org/A014374/b014374.txt", "Table"], {, }][[All, 2]];
etr[f_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d f[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
a = etr[A014374[[# + 1]]&];
a /@ Range[0, Length[A014374] - 1] (* Jean-François Alcover, Dec 04 2019 *)

Euler transformation of A014374.

a(18) from A014374 from Jean-François Alcover, Dec 04 2019

cp's OEIS Frontend

A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

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A185163 Number of not necessarily connected 6-regular simple graphs on n vertices with girth exactly 3.

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A185130 Irregular triangle E(n,g) counting not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g.

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A185336 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 6.

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