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

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..37.

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) = A180867(6,n).

a(n) = 9n(5n-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

Example

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.

Maple

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

Prog

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

Crossrefs

Cf. A033991, A014642, A180579, A180867.

Sequence in context: A066414 A344315 A042422 * A301757 A192328 A129028

Adjacent sequences: A180575 A180576 A180577 * A180579 A180580 A180581

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 30 2010

Status

approved