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A121179 Related to enumeration of alkane systems - see reference for precise definition. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)