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A212744 Number of (w,x,y,z) with all terms in {0,...,n} and w=max{w,x,y,z}-min{w,x,y,z}; i.e., the range of (w,x,y,z) is its first term.

%I A212744 #15 Jan 28 2016 14:31:36
%S A212744 1,8,34,83,181,314,532,791,1177,1604,2206,2843,3709,4598,5776,6959,
%T A212744 8497,10016,11962,13859,16261,18578,21484,24263,27721,31004,35062,
%U A212744 38891,43597,48014,53416,58463,64609,70328,77266,83699,91477,98666,107332,115319,124921
%N A212744 Number of (w,x,y,z) with all terms in {0,...,n} and  w=max{w,x,y,z}-min{w,x,y,z}; i.e., the range of (w,x,y,z) is its first term.
%C A212744 For a guide to related sequences, see A211795.
%H A212744 Colin Barker, <a href="/A212744/b212744.txt">Table of n, a(n) for n = 0..1000</a>
%H A212744 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1)
%F A212744 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 A212744 G.f.: (1+x+x^2)*(x^4+6*x^3+16*x^2+6*x+1) / ((1+x)^3*(x-1)^4).
%F A212744 From _Colin Barker_, Jan 28 2016: (Start)
%F A212744 a(n) = (30*n^3+3*((-1)^n+15)*n^2+3*((-1)^n+15)*n+(-1)^n+15)/16.
%F A212744 a(n) = (15*n^3+24*n^2+24*n+8)/8 for n even.
%F A212744 a(n) = (15*n^3+21*n^2+21*n+7)/8 for n odd.
%F A212744 (End)
%t A212744 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212744 (Do[If[w == Max[w, x, y, z] - Min[w, x, y, z],
%t A212744   s = s + 1],
%t A212744 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212744 Map[t[#] &, Range[0, 40]]   (* A212744 *)
%o A212744 (PARI) Vec((1+x+x^2)*(x^4+6*x^3+16*x^2+6*x+1)/((1+x)^3*(x-1)^4) + O(x^100)) \\ _Colin Barker_, Jan 28 2016
%Y A212744 Cf. A211795.
%K A212744 nonn,easy
%O A212744 0,2
%A A212744 _Clark Kimberling_, May 26 2012