A180577 The Wiener index of the windmill graph D(6,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs).

15, 80, 195, 360, 575, 840, 1155, 1520, 1935, 2400, 2915, 3480, 4095, 4760, 5475, 6240, 7055, 7920, 8835, 9800, 10815, 11880, 12995, 14160, 15375, 16640, 17955, 19320, 20735, 22200, 23715, 25280, 26895, 28560, 30275, 32040, 33855, 35720, 37635, 39600, 41615, 43680, 45795

Offset

1,1

Comments

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

The Wiener polynomial of D(m,n) is (1/2)n(m-1)t[(m-1)(n-1)t+m].

The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1].

For the Wiener indices of D(3,n), D(4,n), and D(5,n) see A033991, A152743, and A028994, respectively.

Links

Table of n, a(n) for n=1..43.

Bruce E. Sagan, Yeong-Nan Yeh, and Ping Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60 (1996), pp. 959-969.

Eric Weisstein's World of Mathematics, Windmill Graph.

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

Formula

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

G.f.: -5*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 30 2012

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

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

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

From Amiram Eldar, Dec 02 2025: (Start)

a(n) = 5 * A147874(n+1).

Sum_{n>=1} 1/a(n) = A294830 / 5. (End)

Maple

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

Prog

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

Crossrefs

Cf. A028994, A033991, A147874, A152743, A294830.

Sequence in context: A189922 A085808 A386257 * A033594 A059377 A370533

Adjacent sequences: A180574 A180575 A180576 * A180578 A180579 A180580

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 21 2010

Extensions

More terms from Elmo R. Oliveira, Apr 03 2025

Status

approved