OFFSET
1,2
COMMENTS
a(n) is the number of hypothetical acyclic hydrocarbons with n carbon atoms that satisfy the octet rule. - Natan Arie Consigli, Dec 26 2016
a(n) is the number of acyclic connected multigraphs with n nodes of degree less than 5, except for a(2). - Natan Arie Consigli, May 25 2017
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
Sean A. Irvine, java program
R. C. Read, Some recent results in chemical enumeration, Preprint, circa 1972. (Annotated scanned copy)
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
Wikipedia, Graph
FORMULA
a(n) ~ c * d^n / n^(5/2), where d = 4.576467424512811226430711636719246756... and c = 0.84315686601314832608482486521039... - Vaclav Kotesovec, Feb 11 2019
EXAMPLE
a(3) = 4 because there are 4 non-cyclic structures that can be drawn with 3 carbons (propane, propene, propyne, and allene). - David Consiglio, Jr., May 15 2014
PROG
(nauty/bash) geng -c -D4 ${n} $[${n}-1]:$[${n}-1] -q | multig -m3 -D4 -u
(PARI) \\ here S is MSET_k comb class of g
S(g, n, k)={polcoeff(exp( sum(i=1, k, (y^i + O(y*y^k))*subst(g + O(x*x^(n\i)), x, x^i)/i )), k, y) + O(x*x^n)}
R(n)={my(f, g, h); f=g=h=O(x); for(n=1, n, h = x*(1+f); g = h + x*(S(f, n, 2) + g); f = g + x*(S(f, n, 3) + f*g + h)); [f, g, h]}
seq(n)={my(t=R(n), f=t[1], g=t[2], h=t[3]); Vec(f + x*(S(f, n, 4) + g*S(f, n, 2) + S(g, n, 2) + f*h) + (subst(f+g+h+O(x*x^(n\2)), x, x^2) - f^2 - g^2 - h^2)/2)} \\ Andrew Howroyd, May 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better definition from Sergio Pimentel, Apr 28 2006
a(11) (computed using Nauty) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
a(12)-a(13) (computed using Molgen 3.5) from David Consiglio, Jr., May 15 2014
Existing terms verified and a(14)-a(16) from Sean A. Irvine, Dec 22 2014
a(17)-a(19) from Sean A. Irvine, Dec 28 2014
a(18)-a(19) corrected and a(20)-a(24) (computed using nauty) from Sean A. Irvine, Jan 02 2015
Terms a(25) and beyond from Andrew Howroyd, May 26 2018
STATUS
approved