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

42, 2697, 29805, 163914, 616008, 1819539, 4550763, 10075380, 20321478, 38078781, 67224201, 112973694, 182160420, 283539207, 428117319, 629511528, 904331490, 1272589425, 1758136101, 2389123122, 3198491520, 4224486651, 5511199395, 7109133660, 9075800190, 11476336677, 14384154177

Offset

1,1

Comments

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).

The Wiener index is in A143366.

The Wiener polynomials for n=1,2,3,4,5 are given in A214581.

Links

Table of n, a(n) for n=1..27.

S. Klavzar, A bird's eye view of the cut method and a survey of its applications in chemical graph theory, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.

Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.

P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (1/10)n +(17/15)n^2 -3n^3 -(55/6)n^4 +(82/5)n^5 +(548/15)n^6.

G.f. = 3*x*(14 +801*x +3936*x^2 +3482*x^3 +530*x^4 +5*x^5)/(1-x)^7.

Example

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.

Maple

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);

Mathematica

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {42, 2697, 29805, 163914, 616008, 1819539, 4550763}, 30] (* Harvey P. Dale, Feb 11 2024 *)

Crossrefs

Cf. A143366, A214581.

Sequence in context: A180371 A046198 A361371 * A294626 A361370 A218409

Adjacent sequences: A214577 A214578 A214579 * A214581 A214582 A214583

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 31 2012

Status

approved