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

%I A224454 #22 Sep 08 2022 08:46:04
%S A224454 27,180,567,1296,2475,4212,6615,9792,13851,18900,25047,32400,41067,
%T A224454 51156,62775,76032,91035,107892,126711,147600,170667,196020,223767,
%U A224454 254016,286875,322452,360855,402192,446571,494100,544887,599040,656667,717876,782775,851472,924075,1000692,1081431,1166400
%N A224454 The Wiener index of the linear phenylene with n hexagons.
%C A224454 a(2) and a(5) have been checked by the direct computation of the Wiener index (using Maple).
%D A224454 I. Gutman, The topological indices of linear phenylenes, J. Serb. Chem. Soc., 60, No. 2, 1995, 99-104.
%H A224454 G. C. Greubel, <a href="/A224454/b224454.txt">Table of n, a(n) for n = 1..1000</a>
%H A224454 G. Cash, S. Klavzar, M. Petkovsek, <a href="http://pubs.acs.org/doi/abs/10.1021/ci0100999">Three methods for calculation of the hyper-Wiener index of a molecular graph</a>, J. Chem. Inf. Comput. Sci. 42, 2002, 571-576.
%H A224454 L. Pavlovic, I. Gutman, <a href="http://pubs.acs.org/doi/abs/10.1021/ci960153z">Wiener numbers of phenylenes: an exact result</a>, J. Chem. Inf. Comput. Sci. 37, 1997, 355-358.
%H A224454 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)
%F A224454 a(n) = 9*n^2*(2n+1) = 9*A099721(n).
%F A224454 G.f.: 9*x*(3 + 8*x + x^2)/(1-x)^4.
%F A224454 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.
%F A224454 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _G. C. Greubel_, Dec 08 2016
%p A224454 a := proc (n) options operator, arrow: 9*n^2*(2*n+1) end proc: seq(a(n), n = 1 .. 40);
%t A224454 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 *)
%o A224454 (PARI) Vec(9*x*(3 + 8*x + x^2)/(1-x)^4 + O(x^50)) \\ _G. C. Greubel_, Dec 08 2016
%o A224454 (Magma) [9*n^2*(2*n+1): n in [1..40]]; // _Vincenzo Librandi_, Dec 09 2016
%Y A224454 Cf. A224455.
%K A224454 nonn,easy
%O A224454 1,1
%A A224454 _Emeric Deutsch_, Apr 10 2013