A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).

27, 144, 351, 648, 1035, 1512, 2079, 2736, 3483, 4320, 5247, 6264, 7371, 8568, 9855, 11232, 12699, 14256, 15903, 17640, 19467, 21384, 23391, 25488, 27675, 29952, 32319, 34776, 37323, 39960, 42687, 45504, 48411, 51408, 54495, 57672, 60939, 64296, 67743, 71280, 74907

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)=27 because in D(6,1)=C_6 we have 6 distances equal to 1, 6 distances equal to 2, and 3 di stances equal to 3.
		

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, A147874, A180579, A180867.

Programs

Maple

seq(9*n*(5*n-2), n = 1 .. 40);

PARI

a(n)=9*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = A180867(6,n).

a(n) = 9*n*(5*n-2).

The Wiener polynomial of the graph D(6,n) is (1/2)nt(t^2+2t+2)((n-1)t^3+2(n-1)t^2+2(n-1)t+6).

G.f.: -9*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 31 2012

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

E.g.f.: 9*exp(x)*x*(3 + 5*x).

a(n) = 9*A147874(n+1).

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

Duplicated a(38) removed by Sean A. Irvine, Apr 14 2025