%I #12 Dec 03 2017 10:12:56
%S 0,0,1,2,3,5,8,10,14,17,22,26,32,36,44,49,57,63,73,79,90,97,109,117,
%T 130,138,153,162,177,187,204,214,232,243,262,274,294,306,328,341,363,
%U 377,401,415,440,455,481,497,524,540,569,586,615,633,664,682,714
%N Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=4y-3x.
%C For a guide to related sequences, see A211422.
%H Colin Barker, <a href="/A211542/b211542.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,-1,-1,-1,0,1).
%F a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9).
%F G.f.: x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)). - _Colin Barker_, Dec 03 2017
%t t[n_] := t[n] = Flatten[Table[2 w + 3 x - 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
%t c[n_] := Count[t[n], 0]
%t t = Table[c[n], {n, 0, 80}] (* A211542 *)
%t FindLinearRecurrence[t]
%t LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1},{0,0,1,2,3,5,8,10,14},57] (* _Ray Chandler_, Aug 02 2015 *)
%o (PARI) concat(vector(2), Vec(x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ _Colin Barker_, Dec 03 2017
%Y Cf. A211422.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Apr 15 2012