Louis V QUINTAS - cp's OEIS frontend

User: Louis V QUINTAS

Louis V QUINTAS has authored 1 sequences.

A291648 a(n) is the number of simple graphs of order n having at most one cycle (such graphs are called "at most unicyclic graphs").

1, 2, 4, 9, 19, 45, 105, 261, 657, 1708, 4498, 12081, 32752, 89792, 247893, 689004, 1924357, 5398587, 15197830, 42917215, 121507597, 344806293, 980423528, 2792741331, 7967842859, 22765631866, 65131178683, 186560990191, 53497417058, 1535637252938

Offset: 1

Louis V QUINTAS, Aug 28 2017

nonn

a(n) = A005195(n) + A236570(n). Proof: Since an at most unicyclic graph is either a forest or a unicyclic graph and since the latter two types of graphs have been enumerated (see A005195, A236570) the enumeration of the at most unicyclic graphs is the sum of the enumeration of the forests and unicyclic graphs, namely, the sum of the sequences A005195 and A236570, where these sequences start for n >= 1, respectively,
1, 2, 3, 6, 10, 20, 37, 76, ...
0, 0, 1, 3,  9, 25, 68 185, ... .

			For n = 4, a(4) = 6 + 3 = 9 and for n = 5, a(5) = 10 + 9 = 19

E. G. DuCasse, L. V. Quintas, and J. M. Zorluoglu, The At Most Unicyclic Random Graph Process, Mathematics Department, Pace University, New York, No. 1 (2017).

Cf. A005195 (number of forests with n unlabeled nodes), A236570 (number of n-node unicyclic graphs).

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User: Louis V QUINTAS

A291648 a(n) is the number of simple graphs of order n having at most one cycle (such graphs are called "at most unicyclic graphs").

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