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A212691 Number of (w,x,y,z) with all terms in {1,...,n} and w+|x-y|<=|x-z|+|y-z|.

%I A212691 #15 Aug 01 2015 10:27:37
%S A212691 0,0,2,8,26,50,100,154,252,348,510,660,902,1118,1456,1750,2200,2584,
%T A212691 3162,3648,4370,4970,5852,6578,7636,8500,9750,10764,12222,13398,15080,
%U A212691 16430,18352,19888,22066,23800,26250,28194,30932,33098,36140
%N A212691 Number of (w,x,y,z) with all terms in {1,...,n} and w+|x-y|<=|x-z|+|y-z|.
%C A212691 Every term is even.
%C A212691 For a guide to related sequences, see A211795.
%H A212691 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, -3, -3, 3, 1, -1).
%F A212691 a(n)=a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
%F A212691 G.f.: 2*x^2*(1+3*x+6*x^2+x^4+3*x^3) / ( (1+x)^3*(x-1)^4 )
%F A212691 a(n) = n*(n-1)*(14*n-7+3*(-1)^n)/24. - _R. J. Mathar_, Jul 01 2013
%t A212691 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212691 (Do[If[w + Abs[x - y] == Abs[x - z] + Abs[y - z],
%t A212691   s = s + 1],
%t A212691 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212691 Map[t[#] &, Range[0, 40]]   (* A212691 *)
%t A212691 %/2 (* integers *)
%t A212691 LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 0, 2, 8, 26, 50, 100}, 40]
%Y A212691 Cf. A211795.
%K A212691 nonn,easy
%O A212691 0,3
%A A212691 _Clark Kimberling_, May 25 2012