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A000621 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers.
(Formerly M0697 N0258)
28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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.
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).
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
EXAMPLE
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 + ...
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 *)
PROG
(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 */
(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 */
CROSSREFS
Sequence in context: A018068 A120400 A217283 * A191317 A218020 A318520
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Additional comments from Bruce Corrigan, Nov 04 2002
Formulae edited by N. J. A. Sloane, Feb 27 2006
STATUS
approved

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Last modified February 20 17:05 EST 2024. Contains 370213 sequences. (Running on oeis4.)