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