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A211616 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -2<=w+x+y<=2.

%I A211616 #10 Dec 04 2017 18:40:25
%S A211616 0,6,42,102,192,312,462,642,852,1092,1362,1662,1992,2352,2742,3162,
%T A211616 3612,4092,4602,5142,5712,6312,6942,7602,8292,9012,9762,10542,11352,
%U A211616 12192,13062,13962,14892,15852,16842,17862,18912,19992,21102,22242,23412,24612,25842
%N A211616 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -2<=w+x+y<=2.
%C A211616 For a guide to related sequences, see A211422.
%H A211616 Colin Barker, <a href="/A211616/b211616.txt">Table of n, a(n) for n = 0..1000</a>
%H A211616 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211616 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
%F A211616 From _Colin Barker_, Dec 04 2017: (Start)
%F A211616 G.f.: 6*x*(1 + 4*x - x^2 + x^3) / (1 - x)^3.
%F A211616 a(n) = 3*(4 - 5*n + 5*n^2) for n>1.
%F A211616 (End)
%t A211616 t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[-2 <= w + x + y <= 2, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t A211616 Map[t[#] &, Range[0, 70]]  (* A211616 *)
%t A211616 %/6                        (* integers *)
%t A211616 FindLinearRecurrence[%]
%t A211616 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A211616 Join[{0, 6},LinearRecurrence[{3, -3, 1},{42, 102, 192},38]] (* _Ray Chandler_, Aug 02 2015 *)
%o A211616 (PARI) concat(0, Vec(6*x*(1 + 4*x - x^2 + x^3) / (1 - x)^3 + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y A211616 Cf. A211422.
%K A211616 nonn,easy
%O A211616 0,2
%A A211616 _Clark Kimberling_, Apr 16 2012