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A212572 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| <= |x-y| + |y-z|.

%I A212572 #12 Dec 07 2015 01:56:18
%S A212572 0,1,14,71,220,533,1094,2015,3416,5449,8270,12071,17044,23421,31430,
%T A212572 41343,53424,67985,85326,105799,129740,157541,189574,226271,268040,
%U A212572 315353,368654,428455,495236,569549,651910,742911,843104,953121,1073550,1205063,1348284
%N A212572 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| <= |x-y| + |y-z|.
%C A212572 For a guide to related sequences, see A211795.
%H A212572 Colin Barker, <a href="/A212572/b212572.txt">Table of n, a(n) for n = 0..1000</a>
%H A212572 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212572 a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).
%F A212572 From _Colin Barker_, Dec 06 2015: (Start)
%F A212572 a(n) = 1/48*(38*n^4+20*n^3-32*n^2-2*(3*(-1)^n-11)*n+3*((-1)^n-1)).
%F A212572 G.f.: x*(1+11*x+30*x^2+26*x^3+9*x^4-x^5) / ((1-x)^5*(1+x)^2).
%F A212572 (End)
%t A212572 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212572 (Do[If[Abs[w - x] <= Abs[x - y] + Abs[y - z], s = s + 1],
%t A212572 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212572 Map[t[#] &, Range[0, 40]]   (* A212572 *)
%o A212572 (PARI) concat(0, Vec(x*(1+11*x+30*x^2+26*x^3+9*x^4-x^5)/((1-x)^5*(1+x)^2) + O(x^100))) \\ _Colin Barker_, Dec 06 2015
%Y A212572 Cf. A211795, A212570.
%K A212572 nonn,easy
%O A212572 0,3
%A A212572 _Clark Kimberling_, May 22 2012