A224455 The hyper-Wiener index of the linear phenylene with n hexagons.

42, 396, 1656, 4740, 10890, 21672, 38976, 65016, 102330, 153780, 222552, 312156, 426426, 569520, 745920, 960432, 1218186, 1524636, 1885560, 2307060, 2795562, 3357816, 4000896, 4732200, 5559450, 6490692, 7534296, 8698956, 9993690, 11427840, 13011072, 14753376, 16665066, 18756780, 21039480

Offset: 1

Author

Emeric Deutsch, Apr 10 2013

Keywords

• nonn
• easy

Comments

a(2) and a(5) have been checked by the direct computation of the hyper-Wiener index (using Maple).

References

• I. Gutman, The topological indices of linear phenylenes, J. Serb. Chem. Soc., 60, No. 2, 1995, 99-104.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• G. Cash, S. Klavzar, M. Petkovsek, Three methods for calculation of the hyper-Wiener index of a molecular graph, J. Chem. Inf. Comput. Sci. 42, 2002, 571-576.
• L. Pavlovic, I. Gutman, Wiener numbers of phenylenes: an exact result, J. Chem. Inf. Comput. Sci. 37, 1997, 355-358.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A224454.

Programs

Maple

a := proc (n) options operator, arrow: (3/2)*n*(n+1)*(9*n^2+3*n+2) end proc: seq(a(n), n = 1 .. 35);

Mathematica

LinearRecurrence[{5,-10,10,-5,1}, {42,396,1656,4740,10890}, 100] (* or *) Table[(3/2)*n*(n+1)*(9*n^2 + 3*n + 2), {n,1,100}] (* G. C. Greubel, Dec 08 2016 *)

PARI

Vec(6*x*(7 + 31*x + 16*x^2)/(1-x)^5 + O(x^50)) \\ G. C. Greubel, Dec 08 2016

Formula

a(n) = (3/2)*n*(n+1)*(9*n^2 + 3*n + 2).

G.f.: 6*x*(7 + 31*x + 16*x^2)/(1-x)^5.

The Hosoya polynomial of the linear phenylene with n hexagons is nt(t^3-t^2-4t-8)/(t-1) + 2t(t+1)(t^(3n)-1)/(t-1)^2.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - G. C. Greubel, Dec 08 2016