A180569 The Wiener index of the P_3 X P_n grid, where P_m is the path graph on m nodes. The Wiener index of a connected graph is the sum of distances between all unordered pairs of nodes in the graph.

4, 25, 72, 154, 280, 459, 700, 1012, 1404, 1885, 2464, 3150, 3952, 4879, 5940, 7144, 8500, 10017, 11704, 13570, 15624, 17875, 20332, 23004, 25900, 29029, 32400, 36022, 39904, 44055, 48484, 53200, 58212, 63529, 69160, 75114, 81400, 88027, 95004

Offset

1,1

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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, Grid Graph.

Formula

a(n) = (1/2)*n*(n+3)*(3n-1).

G.f.: z*(4+9*z-4*z^2)/(1-z)^4.

a(n) = Sum_{k=1..n+1} k*A180568(n,k). - corrected by Andrew Howroyd, May 27 2017

Example

a(1)=4 because in P_3 X P_1 = P_3 there are 2 pairs of nodes at distance 1 and one pair at distance 2.

Maple

seq((1/2)*n*(n+3)*(3*n-1), n = 1 .. 40);

Mathematica

Table[n (n + 3) (3 n - 1)/2, {n, 39}] (* or *)
Rest@ CoefficientList[Series[x (4 + 9 x - 4 x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, May 28 2017 *)

Crossrefs

Row 3 of A143368.

Cf. A180568.

Sequence in context: A281339 A298282 A077205 * A302821 A125309 A273361

Adjacent sequences: A180566 A180567 A180568 * A180570 A180571 A180572

Keyword

nonn

Author

Emeric Deutsch, Sep 28 2010

Status

approved