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A212960 Number of (w,x,y) with all terms in {0,...,n} and |w-x| != |x-y|.

%I A212960 #23 Aug 11 2025 21:43:58
%S A212960 0,4,16,44,92,168,276,424,616,860,1160,1524,1956,2464,3052,3728,4496,
%T A212960 5364,6336,7420,8620,9944,11396,12984,14712,16588,18616,20804,23156,
%U A212960 25680,28380,31264,34336,37604,41072,44748,48636,52744,57076
%N A212960 Number of (w,x,y) with all terms in {0,...,n} and |w-x| != |x-y|.
%C A212960 a(n)+A212959(n)=(n+1)^3.  Every term is divisible by 4.
%C A212960 For a guide to related sequences, see A212959.
%H A212960 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A212960 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
%F A212960 G.f.: f(x)/g(x), where f(x)=4x(1+x^2+x^3) and g(x)=(1+x)(1-x)^4.
%F A212960 a(n) = (4*n^3 + 6*n^2 + 4*n+1 - (-1)^n)/4. - _Luce ETIENNE_, Apr 05 2014
%F A212960 E.g.f.: (x*(7 + 9*x + 2*x^2)*cosh(x) + (1 + 7*x + 9*x^2 + 2*x^3)*sinh(x))/2. - _Stefano Spezia_, Aug 11 2025
%t A212960 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212960 (Do[If[Abs[w - x] != Abs[x - y], s = s + 1],
%t A212960 {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A212960 m = Map[t[#] &, Range[0, 45]]   (* A212960 *)
%t A212960 m/4 (* integers *)
%Y A212960 Cf. A212959.
%K A212960 nonn,easy
%O A212960 0,2
%A A212960 _Clark Kimberling_, Jun 01 2012