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Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=4y-3x.
2

%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