cp's OEIS Frontend

A214580 The hyper-Wiener index of the circumcoronene H(n) (see definition in the Klavzar papers).

%I A214580 #18 Aug 06 2024 10:37:44
%S A214580 42,2697,29805,163914,616008,1819539,4550763,10075380,20321478,
%T A214580 38078781,67224201,112973694,182160420,283539207,428117319,629511528,
%U A214580 904331490,1272589425,1758136101,2389123122,3198491520,4224486651,5511199395,7109133660,9075800190,11476336677,14384154177
%N A214580 The hyper-Wiener index of the circumcoronene H(n) (see definition in the Klavzar papers).
%C A214580 The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).
%C A214580 The Wiener index is in A143366.
%C A214580 The Wiener polynomials for n=1,2,3,4,5 are given in A214581.
%H A214580 S. Klavzar, <a href="https://citeseerx.ist.psu.edu/pdf/17c4b0fe4eaf5d96053952a0ffb5ac0b1fbfbe3d">A bird's eye view of the cut method and a survey of its applications in chemical graph theory</a>, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
%H A214580 Bo-Yin Yang and Yeong-Nan Yeh, <a href="https://doi.org/10.1111/j.0022-2526.2004.01516.x">A Crowning Moment for Wiener Indices</a>, Studies in Appl. Math., 112 (2004), 333-340.
%H A214580 P. Zigert, S. Klavzar, and I. Gutman, <a href="https://www.researchgate.net/publication/289452341_Calculating_the_hyper-Wiener_index_of_benzenoid_hydrocarbons">Calculating the hyper-Wiener index of benzenoid hydrocarbons</a>, ACH Models Chem., 137, 2000, 83-94.
%H A214580 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A214580 a(n) = (1/10)n +(17/15)n^2 -3n^3 -(55/6)n^4 +(82/5)n^5 +(548/15)n^6.
%F A214580 G.f. = 3*x*(14 +801*x +3936*x^2 +3482*x^3 +530*x^4 +5*x^5)/(1-x)^7.
%e A214580 a(1)=42: for n=1 we have a hexagon; the distances are: 1 (6 times), 2 (6 times), 3 (3 times). Then a(1)=(1/2)*(6*1+6*2+3*3+6*1+6*4+3*9)=42.
%p A214580 a := proc (n) options operator, arrow: (1/10)*n+(17/15)*n^2-3*n^3-(55/6)*n^4+(82/5)*n^5+(548/15)*n^6 end proc: seq(a(n), n = 1 .. 30);
%t A214580 LinearRecurrence[{7,-21,35,-35,21,-7,1},{42,2697,29805,163914,616008,1819539,4550763},30] (* _Harvey P. Dale_, Feb 11 2024 *)
%Y A214580 Cf. A143366, A214581.
%K A214580 nonn,easy
%O A214580 1,1
%A A214580 _Emeric Deutsch_, Aug 31 2012