This site is supported by donations to The OEIS Foundation.



The OEIS Foundation is grateful to everyone who made a donation during our Annual Appeal.     Visit the new and spectacular Pictures from the OEIS page!

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000628 Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.
(Formerly M0732 N0274)
1, 1, 1, 1, 2, 3, 5, 11, 24, 55, 136, 345, 900, 2412, 6563, 18127, 50699, 143255, 408429, 1173770, 3396844, 9892302, 28972080, 85289390, 252260276, 749329719, 2234695030, 6688893605, 20089296554, 60526543480, 182896187256, 554188210352, 1683557607211, 5126819371356, 15647855317080, 47862049187447, 146691564302648, 450451875783866, 1385724615285949 (list; graph; refs; listen; history; text; internal format)



Trees are unrooted; nodes are unlabeled and have degree <= 4.

Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A000602 for the analogous sequence when stereoisomers are not counted as different.

Has also been described as steric planted trees (paraffins) with n nodes.


F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.

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.

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.

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. 44.

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).


Table of n, a(n) for n=0..38.

C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (1932), 1538-1545.

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

R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.

Index entries for sequences related to rooted trees

Index entries for sequences related to trees


Blair and Henze give recurrence (see the Maple code).

For even n a(n) = A086194(n) + A086200(n/2), for odd n a(n) = A086194(n).


s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 0 to 60 do s[n+1/4]:=0 od:for n from 0 to 60 do s[n+1/2]:=0 od:for n from 0 to 60 do s[n+3/4]:=0 od:s[ -1]:=0:for n from 1 to 50 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[n-j-k], k=0..n-j), j=1..n))/n od:for n from 0 to 50 do q[n]:=sum(s[i]*s[n-i], i=0..n) od:for n from 0 to 50 do q[n-1/2]:=0 od:for n from 0 to 40 do f:=n->(3*s[n]+2*s[n/2]+q[(n-1)/2]-q[n]+2*sum(s[j]*s[n-3*j-1], j=0..n/3))/4 od:seq(f(n), n=0..38); # the formulas for s[n+1] and f(n) are from eq.(4) and (12), respectively, of the Robinson et al. paper; s[n]=A000625(n), f(n)=A000628(n); q[n] is the convolution of s[n] with itself; # Emeric Deutsch


max = 40; s[0] = s[1] = 1; s[_] = 0; For[n=1, n <= max, n++, s[n+1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[n-j-k], {k, 0, n-j}], {j, 1, n}])/n]; For[n=0, n <= max, n++, q[n] = Sum[s[i]*s[n-i], {i, 0, n}]]; For[n=0, n <= max, n++, q[n-1/2]=0]; f[n_] := (3*s[n] + 2*s[n/2] + q[(n-1)/2] - q[n] + 2*Sum[s[j]*s[n-3*j-1], {j, 0, n/3}])/4; Table[f[n], {n, 0, max}] (* Jean-François Alcover, Dec 29 2014, after Emeric Deutsch *)


Equals A000626 + A000627.

Cf. A000598, A000602, A000625, A010372, A010373, A086194, A086200.

Sequence in context: A176499 A175234 A060696 * A258804 A006888 A009589

Adjacent sequences:  A000625 A000626 A000627 * A000629 A000630 A000631




N. J. A. Sloane


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

More terms from Emeric Deutsch, May 16 2004



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 6 20:58 EST 2016. Contains 268049 sequences.