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

1,1

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)

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

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 *)

Prog

(PARI) Vec(9*x*(3 + 8*x + x^2)/(1-x)^4 + O(x^50)) \\ G. C. Greubel, Dec 08 2016
(Magma) [9*n^2*(2*n+1): n in [1..40]]; // Vincenzo Librandi, Dec 09 2016

Crossrefs

Cf. A224455.

Sequence in context: A083560 A125337 A126495 * A258637 A228463 A000499

Adjacent sequences: A224451 A224452 A224453 * A224455 A224456 A224457

Keyword

nonn,easy

Author

Emeric Deutsch, Apr 10 2013

Status

approved