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A212681 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y|<|y-z|.

%I A212681 #12 Jan 05 2025 05:53:37
%S A212681 0,0,4,24,88,230,504,966,1696,2772,4300,6380,9144,12714,17248,22890,
%T A212681 29824,38216,48276,60192,74200,90510,109384,131054,155808,183900,
%U A212681 215644,251316,291256,335762,385200,439890,500224,566544,639268
%N A212681 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y|<|y-z|.
%C A212681 Also, the number of (w,x,y,z) with all terms in {1,...,n} and |x-y|>|y-z|.
%C A212681 Every term is even.
%C A212681 For a guide to related sequences, see A211795.
%H A212681 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212681 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 A212681 G.f.: (4*x^2 + 12*x^3 + 20*x^4 + 10*x^5 + 2*x^6)/(1 - 3*x + x^2 + 5*x^3 - 5*x^4 - x^5 + 3*x^6 - x^7).
%F A212681 a(n) + A212682(n) = n^4.
%t A212681 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212681 (Do[If[Abs[x - y] < Abs[y - z], s = s + 1],
%t A212681 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212681 Map[t[#] &, Range[0, 40]]   (* A212681 *)
%t A212681 %/2 (* integers *)
%t A212681 LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 0, 4, 24, 88, 230, 504}, 40]
%Y A212681 Cf. A211795, A212682.
%K A212681 nonn,easy
%O A212681 0,3
%A A212681 _Clark Kimberling_, May 24 2012