A000640 Number of paraffins C_n H_{2n-1} XYZ with n carbon atoms. (Formerly M3466 N1410)

0, 1, 4, 13, 42, 131, 402, 1218, 3657, 10899, 32298, 95257, 279844, 819390, 2392392, 6967956, 20250974, 58744089, 170118980, 491913999, 1420493862, 4096940530, 11803172152, 33970257473, 97678027311, 280624328431, 805587723862

Offset

0,3

References

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

N. J. A. Sloane, Table of n, a(n) for n = 0..100

Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules

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)

Formula

G.f.: A(x) = x*A000598(x)/(1-A000642(x))^3.

Maple

# The following Maple commands are taken from the Chyzak web site:
with(combstruct);
gramm_Alkyl:=Alkyl=Prod(Carbon, Set(Alkyl, card<=3)), Carbon=Atom:
specs_Alkyl:=[Alkyl, {gramm_Alkyl}, unlabeled]:
gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon, S1_Alkyl[X], Set(Alkyl, card<=2)), Prod(Prod(Carbon, X), Set(Alkyl, card<=2))), X=Epsilon:
specs_S1_Alkyl:=[S1_Alkyl[X], {gramm_S1_Alkyl, gramm_Alkyl}, unlabeled]:
gramm_S2_Alkyl:=S2_Alkyl[X, Y]=Union(Prod(Carbon, S2_Alkyl[X, Y], Set(Alkyl, card<=2)), Prod(Carbon, Union(S1_Alkyl[X], X), Union(S1_Alkyl[Y], Y), Set(Alkyl, card<=1))):
specs_S2_Alkyl:=[S2_Alkyl[X, Y], {gramm_S2_Alkyl, gramm_S1_Alkyl, op(subs(X=Y, [gramm_S1_Alkyl])), gramm_Alkyl}, unlabeled]:
[seq(count(specs_S2_Alkyl, size=i), i=0..50)];

Mathematica

terms = 27; (* B, B2 = g.f. for A000598, A000642 resp. *) B[_] = 0; Do[B[x_] = 1 + (1/6)*x*(B[x]^3 + 3*B[x]*B[x^2] + 2*B[x^3]) + O[x]^terms // Normal, terms];
B2[x_] = (1/2)*x*(B[x^2] + B[x]^2) + O[x]^terms;
A[x_] = x*B[x]/(1 - B2[x])^3 + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 10 2018 *)

Crossrefs

Cf. A000598, A000642.

Sequence in context: A357063 A307261 A287349 * A199842 A192910 A289807

Adjacent sequences: A000637 A000638 A000639 * A000641 A000642 A000643

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved