A323940 Number of nonisomorphic systems (isomers) for the unsymmetrical schemes (group C_s) for unbranched tri-4-catafusenes as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).

0, 1, 8, 52, 244, 1093, 4490, 17952, 69304, 262385, 973916, 3562532, 12856716, 45880933, 162085694, 567578784, 1971766704, 6801381633, 23309759728, 79421199860, 269160256356, 907726205221, 3047449152562, 10188384019072, 33930769372904

Offset

3,3

Comments

See the comments of sequences A323939, A323941, and A323942 for explanations. - Petros Hadjicostas, May 26 2019

Links

Georg Fischer, Table of n, a(n) for n = 3..103

S. J. Cyvin, B. N. Cyvin and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular Structure 376 (Issues 1-3) (1996), 495-505. See Table 1 on p. 500.

Eric Weisstein's World of Mathematics, Fusene.

Wikipedia, Molecular symmetry.

Wikipedia, Point groups in three dimensions.

Wikipedia, Polyhex (mathematics).

Wikipedia, Schoenflies notation.

Index entries for linear recurrences with constant coefficients, signature (14,-71,116,259,-1246,1013,2520,-5187,594,5931,-4428, -1215,2430,-729).

Formula

a(n) = (1/8) * (1 - (-1)^n) * ((n - 1) - (n + 3) * 3^((n - 5)/2)) + (1/8) * (n^2 + 11 * n + 12) * (n - 2) * 3^(n - 6) - (1/4) * binomial(n, 3) for n >= 3. - Petros Hadjicostas, May 26 2019

G.f.: x^4*(1 -6*x +11*x^2 -32*x^3 +182*x^4 -346*x^5 -122*x^6 +950*x^7 -831*x^8 +336*x^9 -297*x^10 +90*x^11) / ( (1+x)^2*(3*x^2-1)^2*(3*x-1)^4*(x-1)^4 ). - R. J. Mathar, Jul 25 2019

Maple

# Calculates a(r) = AA(r), where r = n is the number of hexagons.
# Crude numbers:
JJ := proc(i) sum(binomial(j + 1, 3)*binomial(i - 2, j - 1)*2^(i - 1 - j), j = 1 .. i - 1); end proc;
# Linearly annelated systems of D_{2h} symmetry:
DD := proc(r) 1/4*(1 - (-1)^r)*(r - 1); end proc;
# Linearly annelated systems of C_{2v} symmetry:
LL := proc(r) 1/2*binomial(r, 3) - (1/8 - 1/8*(-1)^r)*(r - 1); end proc;
# Centrosymmetrical (C_{2h}) systems:
CC := proc(n) 1/24*(1 - (-1)^n)*((3 + n)*3^(1/2*n - 3/2) - 3*n + 3); end proc;
# Total mirror-symmetrical (C_{2v}) systems:
MM := proc(n) CC(n) + LL(n); end proc;
# Unsymmetrical (C_s) systems:
AA := proc(r) 1/4*(JJ(r) - DD(r) - 2*CC(r) - 2*MM(r)); end proc;
# Generate sequence:
for m from 3 to 100 do AA(m); end do; # Petros Hadjicostas, May 26 2019

Mathematica

LinearRecurrence[{14, -71, 116, 259, -1246, 1013, 2520, -5187, 594, 5931, -4428, -1215, 2430, -729}, {0, 1, 8, 52, 244, 1093, 4490, 17952, 69304, 262385, 973916, 3562532, 12856716, 45880933}, 100] (* from the g.f., Georg Fischer, Nov 07 2019 *)

Crossrefs

Cf. A323939, A323941, A323942.

Sequence in context: A244718 A303509 A107584 * A027225 A073377 A055283

Adjacent sequences: A323937 A323938 A323939 * A323941 A323942 A323943

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 09 2019

Extensions

Name edited by Petros Hadjicostas, May 26 2019

More terms using various equations in Cyvin et al. (1996) from Petros Hadjicostas, May 26 2019

Status

approved