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Number of ordered triples (w,x,y) with all terms in {-n, ..., -1, 1, ..., n} and 5w + x + y > 0.
2

%I #13 Dec 06 2017 20:06:01

%S 0,4,32,106,252,495,855,1359,2029,2891,3970,5286,6866,8732,10910,

%T 13425,16297,19553,23215,27309,31860,36888,42420,48478,55088,62275,

%U 70059,78467,87521,97247,107670,118810,130694,143344,156786,171045,186141,202101,218947

%N Number of ordered triples (w,x,y) with all terms in {-n, ..., -1, 1, ..., n} and 5w + x + y > 0.

%C For a guide to related sequences, see A211422.

%H Colin Barker, <a href="/A211630/b211630.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1,-3,3,-1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8) for n > 7.

%F G.f.: x*(4 + 20*x + 22*x^2 + 26*x^3 + 25*x^4 + 16*x^5 + 7*x^6) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Dec 05 2017

%t t = Compile[{{u, _Integer}},

%t Module[{s = 0}, (Do[If[5 w + x + y > 0,

%t s = s + 1], {w, #}, {x, #}, {y, #}] &[

%t Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];

%t Map[t[#] &, Range[0, 60]] (* A211630 *)

%t FindLinearRecurrence[%]

%t (* _Peter J. C. Moses_, Apr 13 2012 *)

%t LinearRecurrence[{3, -3, 1, 0, 1, -3, 3, -1},{0, 4, 32, 106, 252, 495, 855, 1359},36] (* _Ray Chandler_, Aug 02 2015 *)

%o (PARI) concat(0, Vec(x*(4 + 20*x + 22*x^2 + 26*x^3 + 25*x^4 + 16*x^5 + 7*x^6) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ _Colin Barker_, Dec 05 2017

%Y Cf. A211422.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Apr 17 2012