0,5

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.

K. Adam, Title?, MNU 1983, 36, 29 (in German).

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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6.1 Chemical Isomers, p. 299.

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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.

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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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D. H. Rouvray, An introduction to the chemical applications of graph theory, Congress. Numerant., 55 (1986), 267-280. - N. J. A. Sloane, Apr 08 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marten J. ten Hoor, Formula for Success?, Education in Chemistry, 2005, 42(1), 10.

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.

Pontus von Brömssen, Table of n, a(n) for n = 0..1000 (terms n = 0..60 from N. J. A. Sloane)

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

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.

Vijayakumar Ambat, Article about OEIS in Malayalam that mentions this sequence, Mathrubhumi (daily newspaper in Kerala State), circa Nov 20 2022.

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

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 (1988), 304-313.

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

L. Bytautas and D. J. Klein, Chemical combinatorics for alkane-isomer enumeration and more, J. Chem. Inf. Comput. Sci., 38 (1998), 1063-1078.

A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)

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.

R. K. Guy, Letter to N. J. A. Sloane, Mar. 1988

F. Harary and R. Z. Norman, Dissimilarity characteristic theorems for graphs, Proc. Amer. Math. Soc. 11 (1960), 332-334.

H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (8) (1931), 3077-3085.

H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)

H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.

F. Hermann, Ueber das Problem, die Anzahl der isomeren Paraffine der Formel C_n H_{2n+2} zu bestimmung, Chem. Ber., 13 (1880), 792. [I (1880), Annotated scanned copy]

F. Hermann, Ueber das Problem, die Anzahl der isomeren Paraffine der Formel C_n H_{2n+2} zu bestimmung/a>, Chem. Ber., 30 (1897), 2423-2426. [II (1897), Annotated scanned copy]

F. Hermann, Entgegnung, Chem. Ber., 31 (1898), 91. [III (1898), Annotated scanned copy]

Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. Daylight CIS, Inc., EuroMUG '04;4-Nov 05 2004.

E. V. Konstantinova and M. V. Vidyuk, Discriminating tests of information and topological indices; animals and trees; J. Chem. Inf. Comput. Sci., 43 (2003), 1860-1871.

J. Lederberg, Letter to N. J. A. Sloane, Aug 13 1971

J. Lederberg, The topology of molecules, in The Mathematical Sciences. Cambridge, Mass.: MIT Press, 1969, pages 37-51. [Annotated scanned copy]

J. Lederberg, Dendral-64, I, Report to NASA, Dec 1964 [Annotated scanned copy]

J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]

J. Lederberg, Dendral-64, III, Report to NASA, Dec 1968 [Annotated scanned copy]

P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80.

P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

S. M. Losanitsch, Bemerkungen zu der Hermasnnschen Mitteillung: Die Anzahl..., Chem. Ber., 30 (1897), 3059-3060 [Annotated scanned copy]

A. Masoumi, M. Antoniazzi, and M. Soutchanski, Modeling organic chemistry and planning organic synthesis, GCAI 2015. Global Conference on Artificial Intelligence (2015), pp. 176-195.

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D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly] (Annotated scanned copy)

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443. (Annotated scanned copy)

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.

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R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)

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

N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

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

Sylvia Wenmackers, Huiswerk (met bijna twintig jaar vertraging) (in Dutch), 2015.

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 + ...

G.f.: B(x) - cycle_index(S2,-B(x)) + x * cycle_index(S4,B(x)) = B(x) - (B(x)^2 - B(x^2)) / 2 + x * (B(x)^4 + 6*B(x)^2*B(x^2) + 8*B(x)*B(x^3) + 3*B(x^2)^2 + 6*B(x^4)) / 24, where B(x) = 1 + x * cycle_index(S3,B(x)) = 1 + x * (B(x)^3 + 3*B(x)*B(x^2) + 2*B(x^3)) / 6 is the generating function for A000598. - Robert A. Russell, Jan 16 2023

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

1

else

A000022(n)+A000200(n);

end if;

end proc:

n = 40; (* algorithm from Rains and Sloane *)

S3[f_, h_, x_] := f[h, x]^3/6 + f[h, x] f[h, x^2]/2 + f[h, x^3]/3;

S4[f_, h_, x_] := f[h, x]^4/24 + f[h, x]^2 f[h, x^2]/4 + f[h, x] f[h, x^3]/3 + f[h, x^2]^2/8 + f[h, x^4]/4;

T[-1, z_] := 1; T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}].Take[CoefficientList[z^(n+1) + 1 + S3[T, h-1, z]z, z], n+1];

Sum[Take[CoefficientList[z^(n+1) + S4[T, h-1, z]z - S4[T, h-2, z]z - (T[h-1, z] - T[h-2, z]) (T[h-1, z]-1), z], n+1], {h, 1, n/2}] + PadRight[{1, 1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h, z] - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}] (* Robert A. Russell, Sep 15 2018 *)

b[n_, i_, t_, k_] := b[n, i, t, k] = If[i<1, 0, Sum[Binomial[b[i-1, i-1,

k, k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];

b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *) n = 40;

gf[x_] = 1 + Sum[b[j-1, j-1, m, m]x^j, {j, 1, n}]; (* G.f. for A000598 *)

ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];

CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],

{x, 0, n}]], x] (* Robert A. Russell, Jan 19 2023 *)

Column k=4 of A144528.

A000602 = A000022 + A000200 for n>0.

Cf. A000598, A000625, A000628, A067608, A067609, A067610.

Sequence in context: A355190 A208986 A080091 * A034790 A047121 A363307

Adjacent sequences: A000599 A000600 A000601 * A000603 A000604 A000605

nonn,easy,core,nice

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

approved