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A000602 Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.
(Formerly M0718 N0267)
1, 1, 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858, 4347, 10359, 24894, 60523, 148284, 366319, 910726, 2278658, 5731580, 14490245, 36797588, 93839412, 240215803, 617105614, 1590507121, 4111846763, 10660307791, 27711253769 (list; graph; refs; listen; history; text; internal format)



Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.

Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000628 for the analogous sequence with stereoisomers counted.

In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The degree of each node is then <= 4.


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R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.

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D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly - compare first link below]

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N. J. A. Sloane, Table of n, a(n) for n = 0..60

Jean-Fran├žois Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678

R. Aringhieri, P. Hansen and F. Malucelli, Chemical Tree Enumeration Algorithms, Report TR-99-09, Dept. Informatica, Univ. Pisa, 1999.

H. Bottomley, Illustration of initial terms of A000022, A000200, A000602

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478

Alfred W. Francis, Numbers of Isomeric Alkylbenzenes, J. Am. Chem. Soc., 69:6 (1947), pp. 1536-1537.

Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. Daylight CIS, Inc., EuroMUG '04;4-5 November 2004.

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees)., J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678

Stephan Wagner, On the average Wiener index of degree-restricted trees

Index entries for sequences related to trees

Index entries for "core" sequences


a(n) = A010372(n) + A010373(n/2) for n even, a(n) = A010372(n) for n odd.

Also equals A000022 + A000200 (n>0), both of which have known generating functions. Also g.f. = A000678(x)-A000599(x)+A000598(x^2) = (x+x^2+2x^3...)-(x^2+x^3+3x^4...)+(1+x^2+x^4+...) = 1+x+x^2+x^3+2x^4+3x^5...


a(6)=5 because hexane has five isomers: n-hexane; 2-methylpentane; 3-methylpentane; 2,2-dimethylbutane; 2,3-dimethylbutane. - Michael Lugo (mtlugo(AT)mit.edu), Mar 15 2003 (corrected by Andrey V. Kulsha, Sep 22 2011)


A000602 := proc(n) if n=0 then RETURN(1) else A000022(n)+A000200(n); fi; end;


Cf. A000598, A000625, A000628, A067608-A067610.

Sequence in context: A056766 A208986 A080091 * A034790 A047121 A182080

Adjacent sequences:  A000599 A000600 A000601 * A000603 A000604 A000605




N. J. A. Sloane.


Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20, 2003.

Kappler reference from Jonathan Vos Post, Dec 15 2005



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Last modified April 16 15:37 EDT 2014. Contains 240600 sequences.