A060576 a(n) = 1 except for a(1) = 0.

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Vladeta Jovovic, Apr 03 2001

Keywords

• nonn
• easy

Comments

Old name: Number of homeomorphically irreducible general graphs on 1 labeled node and with n edges.

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

This sequence is also produced by Wolfram's Rule 253 of Elementary Cellular Automaton as a triangle read by rows giving successive states initiated with a single ON (black) cell. See the Wolfram, Weisstein and Index links below. - Robert Price, Jan 31 2016

Decimal expansion of 91/900. - Elmo R. Oliveira, May 05 2024

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Links

• V. Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes.
• V. Jovovic, Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges.
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
• S. Wolfram, A New Kind of Science.
• Index to Elementary Cellular Automata
• Index entries for sequences related to cellular automata
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A003514, A060516, A060533-A060537, A060577-A060581.

Programs

Maple

1, 0, seq(1, n=2..200); # Wesley Ivan Hurt, Apr 12 2017

PARI

a(n)=n!=1 \\ Charles R Greathouse IV, Jun 06 2013

Formula

G.f.: (x^2 - x + 1)/(1 - x). a(0)=1, a(1)=0; a(n)=1, n > 1.

E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^(- 1/2)*exp(- x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp(- x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

E.g.f.: e^x - x. - Paul Barry, May 06 2007

a(n) = 1 - binomial(0,n-1). - Arkadiusz Wesolowski, Feb 10 2012

Extensions

Definition simplified by N. J. A. Sloane, Sep 26 2023