A302298 - OEIS

%I #14 Jun 10 2018 12:49:13

%S 0,16,192,1008,3504,9504,21840,44576,83232,145008,239008,376464,

%T 570960,838656,1198512,1672512,2285888,3067344,4049280,5268016,

%U 6764016,8582112,10771728,13387104,16487520,20137520,24407136,29372112,35114128,41721024,49287024,57912960,67706496

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TaxicabMetric.html">Taxicab Metric</a>

%F Conjectures from _Colin Barker_, Apr 08 2018: (Start)

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

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

%F 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 (End)

%t a[n_]:=(1/2)*Sum[Sum[Sum[Sum[

%t Abs[i2-i1] + Abs[j2-j1],

%t {j1,Abs[i1]-n,n-Abs[i1]}],{i1,-n,n}],

%t {j2,Abs[i2]-n,n-Abs[i2]}],{i2,-n,n}];

%t Table[a[n],{n,0,32}]

%Y Cf. A292053, A143945.

%K nonn

%O 0,2

%A _Andres Cicuttin_, Apr 04 2018