A121179 Related to enumeration of alkane systems - see reference for precise definition.

1, 1, 1, 4, 19, 91, 476, 2586, 14421, 82225, 476913, 2804880, 16689036, 100276894, 607588840, 3708251888, 22776251835, 140676848445, 873210347555, 5444307431052, 34080036632565, 214104150405915, 1349504948018208, 8531467913710560, 54083412667272300, 343715994386622918, 2189505804590364876

Offset

0,4

Comments

a(n) is the "number of all staggered conformers of alkyls containing n carbon atoms". It is related to sequence b(n) = A001764(n), which is the number of "space positions of conformers of alkyls related to another alkyl without C_3 symmetry" that contain n carbon atoms. The generating functions of the sequences (a(n): n >= 0) and (b(n): n >= 0), with a(0) = b(0) = 1, appear in some of the papers below. - Petros Hadjicostas, Jul 24 2019

Links

Table of n, a(n) for n=0..26.

S. J. Cyvin, Algebraic solution for the numbers of staggered conformers of alkanes, J. Math. Chem. 17 (1995), 291-293.

S. J. Cyvin, J. Brunvoll, B. N. Cyvin, and W. Lüttke, Enumeration of the staggered conformers of alkanes, Zeitschrift für Naturforschung A 50(9) (1995), 857-863.

S. J. Cyvin, B. N. Cyvin, J. Brumvoll, and Jianji Wang, Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes, J. Molec. Struct. 445 (1998), 127-137.

S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.

Jianji Wang, Shiming Cao, and Ying Li, An algebraic solution for the numbers of staggered conformers of alkanes, J. Math. Chem. 20 (1996), 211-212.

Formula

From Petros Hadjicostas, Jul 24 2019: (Start)

We have a(0) = 1, while for n >= 1 we have

a(n) = (1/3) * A001764(n) = binomial(3*n, n)/(3*(2*n + 1)) if n !== 1 (mod 3), and

a(n) = (1/3) * A001764(n) + (2/3) * A001764((n-1)/3) if n == 1 (mod 3).

G.f.: 1 + (x/3) * (B(x)^3 + 2*B(x^3)), where B(x) is the g.f. of sequence A001764, which satisfies the functional equation B(x) = 1 + x*B^3(x). (It also satisfies the equation B(x) = 1/(1 - x*B^2(x)).) We have

B(x) = (2/sqrt(3*x)) * sin((1/3) * arcsin(sqrt(27*x/4))).

(End)

Maple

A121179 := proc(n)
    if n = 0 then
        return 1;
    elif modp(n, 3) <> 1 then
        A001764(n) ;
    else
        A001764(n)+2*A001764((n-1)/3) ;
    end if;
    %/3 ;
end proc:
seq(A121179(n), n=0..30) ; # R. J. Mathar, Jul 31 2019

Mathematica

b[n_] := Binomial[3n, n]/(2n + 1);
a[n_] := If[n == 0, 1, If[Mod[n, 3] != 1, b[n], b[n] + 2 b[(n-1)/3]]/3];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 01 2020 *)

Crossrefs

Cf. A001764, A121190.

Sequence in context: A004253 A218988 A151253 * A181950 A288687 A275751

Adjacent sequences: A121176 A121177 A121178 * A121180 A121181 A121182

Keyword

nonn

Author

N. J. A. Sloane, Aug 15 2006

Extensions

More terms from Petros Hadjicostas, Jul 24 2019

Status

approved