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A334275 Number of unlabeled connected graphs with n vertices such that every vertex has exactly 2 vertices at distance 2. 0
1, 0, 0, 0, 0, 1, 11, 9, 7, 5, 6, 7, 10, 11, 14, 18, 22, 26, 34, 40, 50, 61, 74, 89, 111, 131, 159, 192, 231, 274, 332, 392, 469, 557, 661, 780, 928, 1088, 1285, 1511, 1776, 2076, 2439, 2843, 3324, 3873, 4511, 5238, 6096, 7057, 8183, 9466 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Gaar and Krenn call these graphs 2-metamour-regular.

LINKS

Table of n, a(n) for n=0..51.

E. Gaar and D. Krenn, Metamour-regular Polyamorous Relationships and Graphs, arXiv:2005.14121 [math.CO], 2020.

FORMULA

a(n) = p_3(n) + 1 for n >= 9 with p_3(n) being the number of integer partitions of n with parts at least 3 (A008483).

EXAMPLE

For n = 8 vertices, there exist the connected 2-metamour-regular graphs

   - c(C_8), c(C_5) join c(C_3), c(C_4) join c(C_4),

   - C_8 and

   - 3 exceptional graphs,

where C_i is the cycle graph on i vertices, and c(G) is the complement graph of G.

Therefore the unlabeled total is a(8) = 7.

PROG

(SageMath) [(len(Partitions(n, min_part=3)) if n >= 6 else 0)

           + (1 if n >= 5 else 0)

           + {0: 1, 6: 8, 7: 6, 8: 3}.get(n, 0)

           for n in srange(52)]

(PARI) a(n)=if(n<9, [1, 0, 0, 0, 0, 1, 11, 9, 7, 5][n+1], numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3)+1) \\ Charles R Greathouse IV, Apr 22 2020

CROSSREFS

Cf. A008483.

Sequence in context: A133236 A038322 A299972 * A090075 A004500 A342162

Adjacent sequences:  A334272 A334273 A334274 * A334276 A334277 A334278

KEYWORD

nonn

AUTHOR

Daniel Krenn, Apr 21 2020

STATUS

approved

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Last modified October 5 21:37 EDT 2022. Contains 357261 sequences. (Running on oeis4.)