A180579 The Wiener index of the Dutch windmill graph D(5,n) (n>=1).

15, 78, 189, 348, 555, 810, 1113, 1464, 1863, 2310, 2805, 3348, 3939, 4578, 5265, 6000, 6783, 7614, 8493, 9420, 10395, 11418, 12489, 13608, 14775, 15990, 17253, 18564, 19923, 21330, 22785, 24288, 25839, 27438, 29085, 30780, 32523, 34314

Offset

1,1

Comments

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

Links

Table of n, a(n) for n=1..38.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Eric Weisstein's World of Mathematics, Dutch Windmill Graph.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 3n(8n-3).

a(n) = A180867(4,n).

The Wiener polynomial of the graph D(5,n) is nt(t+1)[2(n-1)t^2+2(n-1)t+5].

G.f.: -3*x*(11*x+5)/(x-1)^3. - Colin Barker, Oct 31 2012

Example

a(1)=15 because in D(5,1)=C_5 we have 5 distances equal to 1 and 5 distances equal to 2.

Maple

seq(3*n*(8*n-3), n = 1 .. 40);

Mathematica

Table[3n(8n-3), {n, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {15, 78, 189}, 40] (* Harvey P. Dale, May 01 2023 *)

Prog

(PARI) a(n)=3*n*(8*n-3) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A033991, A014642, A180578, A180867.

Sequence in context: A205433 A303097 A128272 * A081591 A269620 A269436

Adjacent sequences: A180576 A180577 A180578 * A180580 A180581 A180582

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 30 2010

Status

approved