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%I #59 May 05 2024 20:03:22
%S 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,
%T 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,
%U 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
%N a(n) = 1 except for a(1) = 0.
%C Old name: Number of homeomorphically irreducible general graphs on 1 labeled node and with n edges.
%C A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.
%C 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
%C Decimal expansion of 91/900. - _Elmo R. Oliveira_, May 05 2024
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
%H V. Jovovic, <a href="/A060576/a060576.pdf">Generating functions for homeomorphically irreducible general graphs on n labeled nodes</a>.
%H V. Jovovic, <a href="/A060576/a060576_rec.pdf">Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>.
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F G.f.: (x^2 - x + 1)/(1 - x). a(0)=1, a(1)=0; a(n)=1, n > 1.
%F 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!.
%F E.g.f.: e^x - x. - _Paul Barry_, May 06 2007
%F a(n) = 1 - binomial(0,n-1). - _Arkadiusz Wesolowski_, Feb 10 2012
%p 1, 0, seq(1, n=2..200); # _Wesley Ivan Hurt_, Apr 12 2017
%o (PARI) a(n)=n!=1 \\ _Charles R Greathouse IV_, Jun 06 2013
%Y Cf. A003514, A060516, A060533-A060537, A060577-A060581.
%K nonn,easy
%O 0,1
%A _Vladeta Jovovic_, Apr 03 2001
%E Definition simplified by _N. J. A. Sloane_, Sep 26 2023
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