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A212900 Number of (w,x,y,z) with all terms in {0,...,n} and distinct consecutive gap sizes.

%I A212900 #17 Jan 05 2025 12:31:29
%S A212900 0,4,28,122,340,786,1558,2814,4690,7404,11130,16140,22652,30992,41416,
%T A212900 54310,69968,88830,111234,137674,168526,204344,245542,292728,346360,
%U A212900 407100,475444,552114,637644,732810,838190,954614,1082698
%N A212900 Number of (w,x,y,z) with all terms in {0,...,n} and distinct consecutive gap sizes.
%C A212900 The gap sizes are |w-x|, |x-y|, |y-z|.  Every term is even.
%C A212900 For a guide to related sequences, see A211795.
%H A212900 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,-1,1,3,-1,-2,1).
%F A212900 a(n) = 2*a(n-1)+a(n-2)-3*a(n-3)-a(n-4)+a(n-5)+3*a(n-6)-a(n-7)-2*a(n-8)+a(n-9).
%F A212900 G.f.: 2*x*(2 + 10*x + 31*x^2 + 40*x^3 + 36*x^4 + 18*x^5 + 7*x^6)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)).
%e A212900 a(1)=4 counts these (w,x,y,z): (0,0,1,1), (0,1,1,0), (1,1,0,0), (1,0,0,1).
%t A212900 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212900 (Do[If[Abs[w - x] != Abs[x - y] && Abs[x - y] != Abs[y - z], s = s + 1],
%t A212900 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212900 m = Map[t[#] &, Range[0, 40]]   (* A212900 *)
%t A212900 m/2  (* integers *)
%t A212900 LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{0,4,28,122,340,786,1558,2814,4690},40] (* _Harvey P. Dale_, Aug 25 2013 *)
%Y A212900 Cf. A211795.
%K A212900 nonn,easy
%O A212900 0,2
%A A212900 _Clark Kimberling_, May 31 2012