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A002986
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Number of non-cyclic hydrocarbons with n carbon atoms (excluding stereoisomers).
(Formerly M2376)
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6
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1, 3, 4, 12, 27, 84, 247, 826, 2777, 9868, 35579, 131847, 495671, 1893819, 7320954, 28619581, 112923053, 449343946, 1801330288, 7269849395, 29517342098, 120507480668, 494449558111, 2038073860257, 8436185990286, 35055744550563, 146195133355612, 611723431211193
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(5/2), where d = 4.576467424512811226430711636719246756... and c = 0.84315686601314832608482486521039... - Vaclav Kotesovec, Feb 11 2019
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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Cf. A000602 (restriction to alkanes).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(11) (computed using Nauty) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
Existing terms verified and a(14)-a(16) from Sean A. Irvine, Dec 22 2014
a(18)-a(19) corrected and a(20)-a(24) (computed using nauty) from Sean A. Irvine, Jan 02 2015
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STATUS
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approved
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