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

%I A302298 #15 Feb 16 2025 08:33:53
%S A302298 0,16,192,1008,3504,9504,21840,44576,83232,145008,239008,376464,
%T A302298 570960,838656,1198512,1672512,2285888,3067344,4049280,5268016,
%U A302298 6764016,8582112,10771728,13387104,16487520,20137520,24407136,29372112,35114128,41721024,49287024,57912960,67706496
%N A302298 Wiener index of the graph of nodes (i,j) of the square lattice such that abs(i) + abs(j) <= n.
%C A302298 The considered grid distance is the Manhattan distance (taxicab metric).
%H A302298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A302298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H A302298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TaxicabMetric.html">Taxicab Metric</a>
%F A302298 Conjectures from _Colin Barker_, Apr 08 2018: (Start)
%F A302298 G.f.: 16*x*(1 + x)*(1 + 5*x + x^2) / (1 - x)^6.
%F A302298 a(n) = 2*(n*(6 + 25*n + 40*n^2 + 35*n^3 + 14*n^4)) / 15.
%F A302298 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.
%F A302298 (End)
%t A302298 a[n_]:=(1/2)*Sum[Sum[Sum[Sum[
%t A302298 Abs[i2-i1] + Abs[j2-j1],
%t A302298 {j1,Abs[i1]-n,n-Abs[i1]}],{i1,-n,n}],
%t A302298 {j2,Abs[i2]-n,n-Abs[i2]}],{i2,-n,n}];
%t A302298 Table[a[n],{n,0,32}]
%Y A302298 Cf. A292053, A143945.
%K A302298 nonn
%O A302298 0,2
%A A302298 _Andres Cicuttin_, Apr 04 2018