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A211440 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w+3x+3y=0.

%I A211440 #15 Aug 29 2021 16:30:51
%S A211440 1,3,5,17,23,29,53,63,73,109,123,137,185,203,221,281,303,325,397,423,
%T A211440 449,533,563,593,689,723,757,865,903,941,1061,1103,1145,1277,1323,
%U A211440 1369,1513,1563,1613,1769,1823,1877,2045,2103,2161,2341,2403,2465
%N A211440 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w+3x+3y=0.
%C A211440 For a guide to related sequences, see A211422.
%H A211440 Robert Israel, <a href="/A211440/b211440.txt">Table of n, a(n) for n = 0..10000</a>
%H A211440 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1)
%F A211440 Conjectures from _Colin Barker_, May 15 2017: (Start)
%F A211440 G.f.: (1 + 3*x + x^2)*(1 - x + 4*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2).
%F A211440 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6.
%F A211440 (End)
%F A211440 From _Robert Israel_, Apr 03 2019: (Start)
%F A211440 a(3*j) = 10*j^2+6*j+1.
%F A211440 a(3*j+1) = 10*j^2 + 10*j + 3.
%F A211440 a(3*j+2) = 10*j^2 + 14*j + 5.
%F A211440 This has the conjectured g.f. and recurrence. (End)
%p A211440 seq(op([10*j^2+6*j+1, 10*j^2 + 10*j + 3, 10*j^2 + 14*j + 5]),j=0..30); # _Robert Israel_, Apr 03 2019
%t A211440 t[n_] := t[n] = Flatten[Table[2 w + 3 x + 3 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
%t A211440 c[n_] := Count[t[n], 0]
%t A211440 t = Table[c[n], {n, 0, 30}]  (* A211440 *)
%t A211440 (t - 1)/2                    (* integers *)
%t A211440 LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,5,17,23,29,53},60] (* _Harvey P. Dale_, Aug 29 2021 *)
%Y A211440 Cf. A211422.
%K A211440 nonn
%O A211440 0,2
%A A211440 _Clark Kimberling_, Apr 11 2012