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A211613 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>1.

%I A211613 #14 Jun 20 2025 15:13:49
%S A211613 0,1,20,78,199,407,726,1180,1793,2589,3592,4826,6315,8083,10154,12552,
%T A211613 15301,18425,21948,25894,30287,35151,40510,46388,52809,59797,67376,
%U A211613 75570,84403,93899,104082,114976,126605,138993,152164,166142,180951,196615,213158
%N A211613 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>1.
%C A211613 For a guide to related sequences, see A211422.
%H A211613 Colin Barker, <a href="/A211613/b211613.txt">Table of n, a(n) for n = 0..1000</a>
%H A211613 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A211613 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
%F A211613 From _Colin Barker_, Dec 04 2017: (Start)
%F A211613 G.f.: x*(1 + 16*x + 4*x^2 + 3*x^3) / (1 - x)^4.
%F A211613 a(n) = (-6 + 9*n - 9*n^2 + 8*n^3)/2 for n > 0. (End)
%F A211613 E.g.f.: 3 + exp(x)*(4*x^3 + 15*x^2/2 + 4*x - 3). - _Stefano Spezia_, Jun 20 2025
%t A211613 t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + x + y > 1, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t A211613 Map[t[#] &, Range[0, 60]]  (* A211613 *)
%t A211613 FindLinearRecurrence[%]
%t A211613 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A211613 Join[{0},LinearRecurrence[{4, -6, 4, -1},{1, 20, 78, 199},35]] (* _Ray Chandler_, Aug 02 2015 *)
%o A211613 (PARI) concat(0, Vec(x*(1 + 16*x + 4*x^2 + 3*x^3) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y A211613 Cf. A211422.
%K A211613 nonn,easy
%O A211613 0,3
%A A211613 _Clark Kimberling_, Apr 16 2012