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Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers.
(Formerly M0697 N0258)
28

%I M0697 N0258 #90 Jun 02 2022 03:29:07

%S 1,1,1,2,3,5,8,14,23,39,65,110,184,310,520,876,1471,2475,4159,6996,

%T 11759,19775,33244,55902,93984,158030,265696,446746,751128,1262940,

%U 2123444,3570318,6002983,10093259,16970431,28533590,47975381,80664329,135626284,228037752

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

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

%C 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).

%C 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.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 300.

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

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

%H Vaclav Kotesovec, <a href="/A000621/b000621.txt">Table of n, a(n) for n = 1..3000</a>. [This replaces an earlier b-file computed by Vincenzo Librandi (and corrected terms 64-1000).]

%H C. M. Blair and H. R. Henze, <a href="http://dx.doi.org/10.1021/ja01342a036">The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins</a>, J. Amer. Chem. Soc., 54 (1932), 1098-1105.

%H C. M. Blair and H. R. Henze, <a href="/A000620/a000620.pdf">The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins</a>, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 283

%H G. Polya, <a href="http://dx.doi.org/10.1524/zkri.1936.93.1.415">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441.

%H G. Polya, <a href="/A000598/a000598_3.pdf">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441. (Annotated scanned copy)

%H G. Polya, <a href="http://dx.doi.org/10.1007/BF02546665">Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen</a>, Acta Mathematica, vol.68, no.1, pp.145-254, (1937). (see pp.151-152).

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

%F 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

%F G.f.: x / (1 - x / (1 - x^2 / (1 - x^4 / (1 - ...)))). - _Michael Somos_, Sep 03 2007

%F From _Joerg Arndt_, Oct 15 2011: (Start)

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

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

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

%F (End)

%F 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

%F 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

%F a(n) = A253190(2*n-1,1). - _R. J. Mathar_, Dec 16 2015

%e 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 + ...

%t 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 *)

%t 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 *)

%o (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 */

%o (Maxima)

%o 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);

%o makelist(T(2*n-1,1),n,1,30); /* _Vladimir Kruchinin_, Mar 18 2015 */

%Y Cf. A000620-A000625, A239804, A239806.

%K nonn,nice,easy

%O 1,4

%A _N. J. A. Sloane_

%E Additional comments from Bruce Corrigan, Nov 04 2002

%E Formulae edited by _N. J. A. Sloane_, Feb 27 2006