cp's OEIS Frontend

A212571 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|<|x-y|+|y-z|.

%I A212571 #14 Mar 14 2025 08:59:45
%S A212571 0,0,8,48,168,428,916,1728,2992,4840,7440,10960,15608,21588,29148,
%T A212571 38528,50016,63888,80472,100080,123080,149820,180708,216128,256528,
%U A212571 302328,354016,412048,476952,549220,629420,718080,815808,923168,1040808,1169328,1309416
%N A212571 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|<|x-y|+|y-z|.
%C A212571 For a guide to related sequences, see A211795.
%H A212571 Colin Barker, <a href="/A212571/b212571.txt">Table of n, a(n) for n = 0..1000</a>
%H A212571 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212571 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 A212571 From _Colin Barker_, Dec 05 2015: (Start)
%F A212571 a(n) = 1/48*(38*n^4-20*n^3-32*n^2+2*(3*(-1)^n+13)*n+3*((-1)^n-1)).
%F A212571 G.f.: 4*x^2*(2+6*x+8*x^2+3*x^3) / ((1-x)^5*(1+x)^2). (End)
%F A212571 E.g.f.: (x*(3 + 87*x + 104*x^2 + 19*x^3)*cosh(x) - (3 - 9*x - 87*x^2 - 104*x^3 - 19*x^4)*sinh(x))/24. - _Stefano Spezia_, Mar 12 2025
%t A212571 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212571 (Do[If[Abs[w - x] < Abs[x - y] + Abs[y - z], s = s + 1],
%t A212571 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212571 Map[t[#] &, Range[0, 40]]   (* A212571 *)
%t A212571 %/4  (* integers *)
%t A212571 LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,8,48,168,428,916},40] (* _Harvey P. Dale_, Aug 28 2024 *)
%o A212571 (PARI) concat([0,0], Vec(4*x^2*(2+6*x+8*x^2+3*x^3)/((1-x)^5*(1+x)^2) + O(x^100))) \\ _Colin Barker_, Dec 05 2015
%Y A212571 Cf. A211795, A212570.
%K A212571 nonn,easy
%O A212571 0,3
%A A212571 _Clark Kimberling_, May 22 2012