A224454 The Wiener index of the linear phenylene with n hexagons.

27, 180, 567, 1296, 2475, 4212, 6615, 9792, 13851, 18900, 25047, 32400, 41067, 51156, 62775, 76032, 91035, 107892, 126711, 147600, 170667, 196020, 223767, 254016, 286875, 322452, 360855, 402192, 446571, 494100, 544887, 599040, 656667, 717876, 782775, 851472, 924075, 1000692, 1081431, 1166400

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 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 (4,-6,4,-1)

Crossrefs

Cf. A224455.

Programs

Magma

[9*n^2*(2*n+1): n in [1..40]]; // Vincenzo Librandi, Dec 09 2016

Maple

a := proc (n) options operator, arrow: 9*n^2*(2*n+1) end proc: seq(a(n), n = 1 .. 40);

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {27, 180, 567, 1296}, 100] (* or *) Table[9*n^2*(2*n+1), {n,1,100}] (* G. C. Greubel, Dec 08 2016 *)

PARI

Vec(9*x*(3 + 8*x + x^2)/(1-x)^4 + O(x^50)) \\ G. C. Greubel, Dec 08 2016

Formula

a(n) = 9*n^2*(2n+1) = 9*A099721(n).

G.f.: 9*x*(3 + 8*x + x^2)/(1-x)^4.

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) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - G. C. Greubel, Dec 08 2016