A185236 -id:A185236 - cp's OEIS frontend

A185226 Number of disconnected 2-regular simple graphs on n vertices with girth at least 6.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 11, 15, 16, 20, 23, 28, 31, 39, 43, 52, 59, 70, 79, 95, 106, 125, 142, 166, 187, 220, 247, 287, 325, 375, 423, 490, 551, 633, 715, 818, 921, 1055, 1186, 1352, 1522, 1729, 1943, 2208

Offset: 0

Jason Kimberley, Feb 22 2011

Number of partitions of n with each part at least 6, and at least 2 parts.

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Disconnected k-regular simple graphs with girth at least 6: A185216 (all k), A185206 (triangle); this sequence (k=2), A185236 (k=3), A185246 (k=4).

Disconnected 2-regular simple graphs with girth at least g: A165652 (g=3), A185224 (g=4), A185225 (g=5), this sequence (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).

Magma
```
A185226 := func;
```

a(n) = A185326(n) - A185116(n).

A185216 Number of disconnected regular simple graphs on n vertices with girth at least 6.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 6, 9, 9, 12, 12, 17, 17, 22, 24, 31, 32, 42, 44, 60, 60, 109, 80, 529, 107, 8246, 143, 191422, 188, 4856141, 248, 127938143, 326, 3482858640, 424, 98176518751, 552

Offset: 0

Jason Kimberley, Jun 21 2012

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Disconnected k-regular simple graphs with girth at least 6: this sequence (all k), A185206 (triangle); A185226 (k=2), A185236 (k=3), A185246 (k=4).

Disconnected regular graphs with girth at least g: A068932 (g=3), A185214 (g=4), A185215 (g=5), this sequence (g=6), A185217 (g=7).

A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 23, 0, 1301, 25, 495379, 13529

Offset: 0

Jason Kimberley, Feb 22 2011

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

4-regular simple graphs with girth at least 4: A058348 (connected), this sequence (disconnected), A185346 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), A185244 (g=4), A185245 (g=5), this sequence (g=6).
Disconnected k-regular simple graphs with girth at least 6: A185216 (all k), A185206 (triangle); A185226 (k=2), A185236 (k=3), this sequence (k=4).

a(n) = A185346(n) - A058348(n) = Euler_transformation(A058348)(n) - A058348(n).

A185035 Number of disconnected 3-regular simple graphs on 2n vertices with girth exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 12, 67, 597, 7134, 107820, 1876672, 35924730, 741405102, 16356067055, 383931363314

Offset: 0

Jason Kimberley, Feb 29 2012

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

Disconnected 3-regular simple graphs with girth exactly g: A185033 (g=3), A185034 (g=4), this sequence (g=5), A185036 (g=6), A185037 (g=7).

a(n) = A185235(n) - A185236(n).

A185036 Number of disconnected 3-regular simple graphs on 2n vertices with girth exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 37, 432, 8119, 191254, 4855919, 127937854, 3482858263, 98176518258

Offset: 0

Jason Kimberley, Feb 29 2012

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

Disconnected 3-regular simple graphs with girth exactly g: A185033 (g=3), A185034 (g=4), A185035 (g=5), this sequence (g=6), A185037 (g=7).

a(n) = A185236(n) - A185237(n).

A185206 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth at least 6.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 3, 1, 1, 5, 1, 0, 5, 1, 1, 7, 1, 0, 8, 1, 1, 10, 1, 0, 11, 1, 1, 15, 1, 0, 16, 1, 1, 20, 1, 0, 23, 1, 1, 28, 1, 1, 0, 31, 0, 1, 1, 39, 1, 1, 0, 43, 0, 1, 1, 52, 6, 1, 0, 59, 0, 1, 1, 70, 37, 1, 0, 79, 0

Offset: 1

Jason Kimberley, Nov 03 2012

			1: 0;
2: 1;
3: 1;
4: 1, 1;
5: 1, 0;
6: 1, 1;
7: 1, 0;
8: 1, 1;
9: 1, 0;
10: 1, 1;
11: 1, 0;
12: 1, 1, 1;
13: 1, 0, 1;
14: 1, 1, 2;
15: 1, 0, 2;
16: 1, 1, 3;
17: 1, 0, 3;
18: 1, 1, 5;
19: 1, 0, 5;
20: 1, 1, 7;
21: 1, 0, 8;
22: 1, 1, 10;
23: 1, 0, 11;
24: 1, 1, 15;
25: 1, 0, 16;
26: 1, 1, 20;
27: 1, 0, 23;
28: 1, 1, 28, 1;
29: 1, 0, 31, 0;
30: 1, 1, 39, 1;
31: 1, 0, 43, 0;
32: 1, 1, 52, 6;
33: 1, 0, 59, 0;
34: 1, 1, 70, 37;
35: 1, 0, 79, 0;
36: 1, 1, 95, 432;
37: 1, 0, 106, 0;
38: 1, 1, 125, 8119;
39: 1, 0, 142, 0;
40: 1, 1, 166, 191254;
41: 1, 0, 187, 0;
42: 1, 1, 220, 4855919;
43: 1, 0, 247, 0;
44: 1, 1, 287, 127937854;
45: 1, 0, 325, 0;
46: 1, 1, 375, 3482858263;
47: 1, 0, 423, 0;
48: 1, 1, 490, 98176518259;
49: 1, 0, 551, 0;

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Disconnected k-regular simple graphs with girth at least 6: A185216 (all k), this sequence (triangle); A185226 (k=2), A185236 (k=3), A185246 (k=4).

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

A185226 Number of disconnected 2-regular simple graphs on n vertices with girth at least 6.

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A185216 Number of disconnected regular simple graphs on n vertices with girth at least 6.

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A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 6.

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A185035 Number of disconnected 3-regular simple graphs on 2n vertices with girth exactly 5.

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A185036 Number of disconnected 3-regular simple graphs on 2n vertices with girth exactly 6.

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A185206 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth at least 6.

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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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