A302298 Wiener index of the graph of nodes (i,j) of the square lattice such that abs(i) + abs(j) <= n.

0, 16, 192, 1008, 3504, 9504, 21840, 44576, 83232, 145008, 239008, 376464, 570960, 838656, 1198512, 1672512, 2285888, 3067344, 4049280, 5268016, 6764016, 8582112, 10771728, 13387104, 16487520, 20137520, 24407136, 29372112, 35114128, 41721024, 49287024, 57912960, 67706496

Offset: 0

Author

Andres Cicuttin, Apr 04 2018

Keywords

• nonn

Comments

The considered grid distance is the Manhattan distance (taxicab metric).

Links

• Eric Weisstein's World of Mathematics, Grid Graph
• Eric Weisstein's World of Mathematics, Wiener Index
• Eric Weisstein's World of Mathematics, Taxicab Metric

Crossrefs

Cf. A292053, A143945.

Programs

Mathematica

a[n_]:=(1/2)*Sum[Sum[Sum[Sum[
Abs[i2-i1] + Abs[j2-j1],
{j1,Abs[i1]-n,n-Abs[i1]}],{i1,-n,n}],
{j2,Abs[i2]-n,n-Abs[i2]}],{i2,-n,n}];
Table[a[n],{n,0,32}]

Formula

Conjectures from Colin Barker, Apr 08 2018: (Start)

G.f.: 16*x*(1 + x)*(1 + 5*x + x^2) / (1 - x)^6.

a(n) = 2*(n*(6 + 25*n + 40*n^2 + 35*n^3 + 14*n^4)) / 15.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.

(End)