A000621 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers.

1, 1, 1, 2, 3, 5, 8, 14, 23, 39, 65, 110, 184, 310, 520, 876, 1471, 2475, 4159, 6996, 11759, 19775, 33244, 55902, 93984, 158030, 265696, 446746, 751128, 1262940, 2123444, 3570318, 6002983, 10093259, 16970431, 28533590, 47975381, 80664329, 135626284, 228037752

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are not stereoisomers.

Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).

Then Ps (and As) = A000620, Pn (and An, Sn) = this sequence, Ss = A000622, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 14*x^8 + 23*x^9 + ...
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 300.
• 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).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..3000. [This replaces an earlier b-file computed by Vincenzo Librandi (and corrected terms 64-1000).]
• C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (1932), 1098-1105.
• C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 283
• G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441.
• G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441. (Annotated scanned copy)
• G. Polya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, vol.68, no.1, pp.145-254, (1937). (see pp.151-152).

Crossrefs

Cf. A000620-A000625, A239804, A239806.

Programs

Mathematica

nmax=40; a=1-x; Do[a=1/(1-x (a/.x->x^2)),{Log[2,nmax]+2}]; CoefficientList[Series[a,{x,0,nmax-1}],x] (* Jean-François Alcover, Jun 16 2011, after Michael Somos, fixed by Vaclav Kotesovec, Mar 28 2014 *)
max = 40; cf = Fold[Function[1 - x^#2/#1], 1, 2^Reverse[Range[0, Floor[Log[2, max]]]]]; List @@ (1-Series[cf, {x, 0, 2*max}] // Normal) /. x -> 1 (* Jean-François Alcover, Sep 24 2014 *)

Maxima

T(n,m):=if m=n then 1 else sum(binomial(i+m-1,i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
makelist(T(2*n-1,1),n,1,30); /* Vladimir Kruchinin, Mar 18 2015 */

PARI

{a(n) = my(A, m); if( n<1, 0, n--; m = 1; A = 1 + O(x); while( m<=n, m *= 2; A = 1 / (1 - x * subst(A, x, x^2)) ); polcoeff( A, n )) }; /* Michael Somos, Sep 03 2007 */

Formula

G.f.: A(x) satisfies A(x) = 1/(1-x*A(x^2)), with offset 0. - Paul D. Hanna, Aug 16 2002

Given g.f. A(x), then B(x) = A(x) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (1 - u)^2 * w - u^2 * v * (v - 1). - Michael Somos, Sep 03 2007

G.f.: x / (1 - x / (1 - x^2 / (1 - x^4 / (1 - ...)))). - Michael Somos, Sep 03 2007

From Joerg Arndt, Oct 15 2011: (Start)

For offset 0 (as considered in the 1937 Polya reference) we have

G.f.: A(x) = 1 / (1 - x / (1 - x^2 / (1 - x^4 / (1 - ...)))) and

A(x) satisfies A(x) = 1 + x*A(x)*A(x^2) (equivalent to Hanna's functional equation).

(End)

a(n) ~ c * beta^n, where beta = 1.681367524441880255591... (see A239804), c = 0.214536139134648555630... (see A239806). Asymptotic formula a(n) ~ K * beta^n from reference (Analytic Combinatorics, p. 283), where K = 0.3607140971, beta = 1.6813675244^n is for offset 0 (beta is same, but K = c * beta = 0.360714097160142828...). - Vaclav Kotesovec, Mar 27 2014

a(n) = T(2*n-1,1), where T(n,m) = Sum_{i=1..n-m} binomial(i+m-1,i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i), n > m, T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015

a(n) = A253190(2*n-1,1). - R. J. Mathar, Dec 16 2015

Extensions

Additional comments from Bruce Corrigan, Nov 04 2002

Formulae edited by N. J. A. Sloane, Feb 27 2006