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A212901 Number of (w,x,y,z) with all terms in {0,...,n} and equal consecutive gap sizes.

%I A212901 #13 Jan 05 2025 05:53:30
%S A212901 1,4,13,26,45,66,95,126,163,204,251,300,357,416,481,550,625,702,787,
%T A212901 874,967,1064,1167,1272,1385,1500,1621,1746,1877,2010,2151,2294,2443,
%U A212901 2596,2755,2916,3085,3256,3433,3614,3801,3990,4187,4386,4591
%N A212901 Number of (w,x,y,z) with all terms in {0,...,n} and equal consecutive gap sizes.
%C A212901 The gap sizes are |w-x|, |x-y|, |y-z|.  For a guide to related sequences, see A211795.
%H A212901 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A212901 a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6).
%F A212901 G.f.: f(x)/g(x), where f(x) = 1 + 3*x + 8*x^2 + 9*x^3 + 7*x^4 and g(x) = (1 + 2*x + 2*x^2 + x^3)(1 - x)^3.
%e A212901 a(1)=4 counts these (w,x,y,z): (0,0,0,0), (1,1,1,1), (0,1,0,1), (1,0,1,0).
%t A212901 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212901 (Do[If[Abs[w - x] == Abs[x - y] == Abs[y - z], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212901 m = Map[t[#] &, Range[0, 40]]   (* A212901 *)
%Y A212901 Cf. A211795, A212900.
%K A212901 nonn,easy
%O A212901 0,2
%A A212901 _Clark Kimberling_, May 31 2012