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, 36153, 38040, 39975, 41958, 43989

Offset: 1

Author

Emeric Deutsch, Sep 30 2010

Keywords

• nonn
• easy

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.

Examples

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

Links

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
• Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A014642, A033991, A139273, A180578, A180867.

Programs

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 *)

PARI

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

Formula

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

From Elmo R. Oliveira, Apr 03 2025: (Start)

E.g.f.: 3*exp(x)*x*(5 + 8*x).

a(n) = 3*A139273(n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Extensions

More terms from Elmo R. Oliveira, Apr 03 2025