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%I #21 Dec 04 2017 09:20:48
%S 0,4,29,99,238,470,819,1309,1964,2808,3865,5159,6714,8554,10703,13185,
%T 16024,19244,22869,26923,31430,36414,41899,47909,54468,61600,69329,
%U 77679,86674,96338,106695,117769,129584,142164,155533,169715,184734,200614,217379
%N Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>0.
%C For a guide to related sequences, see A211422.
%H Colin Barker, <a href="/A211545/b211545.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F From _Colin Barker_, Dec 04 2017: (Start)
%F G.f.: x*(4 + 13*x + 7*x^2) / (1 - x)^4.
%F a(n) = (n*(3 - 3*n + 8*n^2))/2.
%F (End)
%e a(1) counts these triples: (-1,1,1), (1,-1,1), (1,1,-1), (1,1,1).
%t t = Compile[{{u, _Integer}},
%t Module[{s = 0}, (Do[If[w + x + y > 0, s = s + 1],
%t {w, #}, {x, #}, {y, #}] &[
%t Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t Map[t[#] &, Range[0, 60]] (* A211545 *)
%t FindLinearRecurrence[%]
%t (* _Peter J. C. Moses_, Apr 13 2012 *)
%t LinearRecurrence[{4,-6,4,-1},{0,4,29,99},36] (* _Ray Chandler_, Aug 02 2015 *)
%o (PARI) concat(0, Vec(x*(4 + 13*x + 7*x^2) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y Cf. A211422.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Apr 16 2012
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