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A211617 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 2w+x+y>0.

%I A211617 #11 Dec 04 2017 11:42:04
%S A211617 0,3,30,101,244,479,834,1329,1992,2843,3910,5213,6780,8631,10794,
%T A211617 13289,16144,19379,23022,27093,31620,36623,42130,48161,54744,61899,
%U A211617 69654,78029,87052,96743,107130,118233,130080,142691,156094,170309,185364,201279,218082
%N A211617 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 2w+x+y>0.
%C A211617 For a guide to related sequences, see A211422.
%H A211617 Colin Barker, <a href="/A211617/b211617.txt">Table of n, a(n) for n = 0..1000</a>
%H A211617 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A211617 a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>4.
%F A211617 From _Colin Barker_, Dec 04 2017: (Start)
%F A211617 G.f.: x*(3 + 21*x + 17*x^2 + 7*x^3) / ((1 - x)^4*(1 + x)).
%F A211617 a(n) = 4*n^3 - n^2 + n for n even.
%F A211617 a(n) = 4*n^3 - n^2 + n - 1 for n odd.
%F A211617 (End)
%t A211617 t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[2 w + x + y > 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[ Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t A211617 Map[t[#] &, Range[0, 70]]  (* A211617 *)
%t A211617 FindLinearRecurrence[%]
%t A211617 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A211617 LinearRecurrence[{3, -2, -2, 3, -1},{0, 3, 30, 101, 244},36] (* _Ray Chandler_, Aug 02 2015 *)
%o A211617 (PARI) concat(0, Vec(x*(3 + 21*x + 17*x^2 + 7*x^3) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y A211617 Cf. A211422.
%K A211617 nonn,easy
%O A211617 0,2
%A A211617 _Clark Kimberling_, Apr 16 2012