%I A000602 M0718 N0267
%S A000602 1,1,1,1,2,3,5,9,18,35,75,159,355,802,1858,4347,10359,24894,60523,
%T A000602 148284,366319,910726,2278658,5731580,14490245,36797588,93839412,
%U A000602 240215803,617105614,1590507121,4111846763,10660307791,27711253769
%N A000602 Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) 
               ignoring stereoisomers.
%C A000602 Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.
%C A000602 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.
%C A000602 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.
%D A000602 K. Adam, Title?, MNU 1983, 36, 29 (in German).
%D A000602 M. van Almsick, H. Dolhaine and H. Honig, Efficient algorithms to enumerate 
               isomers and diamutamers with more than one type of substituent, J. 
               Chem. Info. and Computer Science, 40 (2000), 956-966.
%D A000602 A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes 
               having n carbons and a longest chain of length d, J. Chem. Education, 
               65 (No. 4, 1988), 304-313.
%D A000602 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Camb. 1998, p. 290.
%D A000602 L. Bytautas and D. J. Klein, Chemical combinatorics for alkane-isomer 
               enumeration and more, J. Chem. Inf. Comput. Sci., 38 (1998), 1063-1078.
%D A000602 A. Cayley, Ueber die analytischen Figuren, welche in der Mathematik Baeume 
               genannt werden..., Chem. Ber. 8 (1875), 1056-1059.
%D A000602 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.
%D A000602 J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews 
               in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 
               2005.
%D A000602 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 529.
%D A000602 Handbook of Combinatorics, North-Holland '95, p. 1963.
%D A000602 F. Harary and R. Z. Norman, Dissimilarity characteristic theorems for 
               graphs, Proc. Amer. Math. Soc. 11 (1960), 332-334.
%D A000602 J. B. Hendrickson and C. A. Parks, "Generation and Enumeration of Carbon 
               skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp. 101-107. 
               See Table 2, column 2 on page 103.
%D A000602 H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the 
               methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
%D A000602 H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons 
               of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-686.
%D A000602 M. D. Jackson and T. I. Bieber, Applications of degree distribution, 
               2: construction and enumeration of isomers in the alkane series, 
               J. Chem. Info. and Computer Science, 33 (1993), 701-708.
%D A000602 E. V. Konstantinova and M. V. Vidyuk, Discriminating tests of information 
               and toplogical indices; animals and trees; J. Chem. Inf. Comput. 
               Sci., 43 (2003), 1860-1871.
%D A000602 J. Lederberg et al., Applications of artificial intelligence for chemical 
               systems, I: The number of possible organic compounds. Acyclic structures 
               containing C, H, O and N, J. Amer. Chem. Soc., 91 (1969), 2973-2097.
%D A000602 P. Leroux and B. Miloudi, ``G\'{e}n\'{e}ralisations de la formule d'Otter,
               '' Ann. Sci. Math. Qu\'{e}bec, Vol. 16, No. 1, pp. 53-80, 1992.
%D A000602 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%D A000602 L. M. Masinter, Applications of artificial intelligence for chemical 
               systems, XX, Exhaustive generation of cyclic and acyclic isomers, 
               J. Amer. Chem. Soc., 96 (1974), 7702-7714.
%D A000602 W. R. Mueller et al., Molecular topological index, J. Chem. Inf. Comput. 
               Sci., 30 (1990),160-163.
%D A000602 D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 
               (1932), 2918-2920. [Gives a(60) correctly - compare first link below]
%D A000602 M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical 
               trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 
               563-567.
%D A000602 G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer 
               Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443.
%D A000602 R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 
               of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. 
               Press, 1976; see p. 28.
%D A000602 R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral 
               alkanes..., Tetrahedron 32 (1976), 355-361.
%D A000602 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000602 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000602 Marten J. ten Hoor, Formula for Success?, Education in Chemistry, 2005, 
               42(1), 10.
%D A000602 N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, 
               COMPUTER GENERATION OF ISOMERIC STRUCTURES, Pure & Appl. Chem., Vol. 
               55, No. 2, pp. 379-39O, 1983.
%D A000602 S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic 
               approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. 
               Univ. Graz, Austria, Feb., 2006.
%H A000602 N. J. A. Sloane, <a href="b000602.txt">Table of n, a(n) for n = 0..60</
               a>
%H A000602 R. Aringhieri, P. Hansen and F. Malucelli, <a href="http://citeseer.ist.psu.edu/
               aringhieri99chemical.html">Chemical Tree Enumeration Algorithms</
               a>, Report TR-99-09, Dept. Informatica, Univ. Pisa, 1999.
%H A000602 H. Bottomley, <a href="a602.gif">Illustration of initial terms of A000022, 
               A000200, A000602</a>
%H A000602 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               478
%H A000602 Michael A. Kappler, <a href="http://www.daylight.com/meetings/emug04/
               Kappler/GenSmi.html">GENSMI: Exhaustive Enumeration of Simple Graphs.</
               a> Daylight CIS, Inc., EuroMUG '04;4-5 Nov, 2004.
%H A000602 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A000602 N. J. A. Sloane, <a href="a000602.txt">Maple program and first 60 terms 
               for A000022, A000200, A000598, A000602, A000678</a>
%H A000602 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A000602 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000602 a(n) = A010372(n) + A010373(n/2) for n even, a(n) = A010372(n) for n 
               odd.
%F A000602 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...
%e A000602 a(6)=5 because hexane has five isomers: n-hexane, 2-methylhexane, 3-methylhexane, 
               2,2-dimethylhexane, 2,3-dimethylhexane. - Michael Lugo (mtlugo(AT)mit.edu), 
               Mar 15 2003
%p A000602 A000602 := proc(n) if n=0 then RETURN(1) else A000022(n)+A000200(n); 
               fi; end;
%Y A000602 Cf. A000598, A000625, A000628, A067608-A067610.
%Y A000602 Sequence in context: A047031 A056766 A080091 this_sequence A034790 A047121 
               A096753
%Y A000602 Adjacent sequences: A000599 A000600 A000601 this_sequence A000603 A000604 
               A000605
%K A000602 nonn,easy,core,nice
%O A000602 0,5
%A A000602 N. J. A. Sloane (njas(AT)research.att.com).
%E A000602 Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 
               20, 2003.
%E A000602 Kappler reference from Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 
               15 2005