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A211620 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=2w+x+y<=1.

%I A211620 #11 Dec 04 2017 15:30:24
%S A211620 0,2,16,38,76,122,184,254,340,434,544,662,796,938,1096,1262,1444,1634,
%T A211620 1840,2054,2284,2522,2776,3038,3316,3602,3904,4214,4540,4874,5224,
%U A211620 5582,5956,6338,6736,7142,7564,7994,8440,8894,9364,9842,10336,10838,11356,11882
%N A211620 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=2w+x+y<=1.
%C A211620 For a guide to related sequences, see A211422.
%H A211620 Colin Barker, <a href="/A211620/b211620.txt">Table of n, a(n) for n = 0..1000</a>
%H A211620 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A211620 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
%F A211620 From _Colin Barker_, Dec 04 2017: (Start)
%F A211620 G.f.: 2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)).
%F A211620 a(n) = 6*n^2 - 6*n + 4 for n>0 and even.
%F A211620 a(n) = 6*n^2 - 6*n + 2 for n odd.
%F A211620 (End)
%t A211620 t = Compile[{{u, _Integer}},
%t A211620    Module[{s = 0}, (Do[If[-1 <= 2 w + x + y <= 1,
%t A211620          s = s + 1], {w, #}, {x, #}, {y, #}] &[
%t A211620       Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
%t A211620 Map[t[#] &, Range[0, 70]]  (* A211620 *)
%t A211620 %/2                        (* integers *)
%t A211620 FindLinearRecurrence[%]
%t A211620 (* _Peter J. C. Moses_, Apr 13 2012 *)
%t A211620 Join[{0},LinearRecurrence[{2, 0, -2, 1},{2, 16, 38, 76},42]] (* _Ray Chandler_, Aug 02 2015 *)
%o A211620 (PARI) concat(0, Vec(2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ _Colin Barker_, Dec 04 2017
%Y A211620 Cf. A211422.
%K A211620 nonn,easy
%O A211620 0,2
%A A211620 _Clark Kimberling_, Apr 16 2012