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

%I #11 Dec 04 2017 09:05:20

%S 0,1,11,57,160,344,633,1051,1622,2370,3319,4493,5916,7612,9605,11919,

%T 14578,17606,21027,24865,29144,33888,39121,44867,51150,57994,65423,

%U 73461,82132,91460,101469,112183,123626,135822,148795,162569,177168,192616,208937

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

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

%H Colin Barker, <a href="/A211614/b211614.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>5.

%F From _Colin Barker_, Dec 04 2017: (Start)

%F G.f.: x*(1 + 7*x + 19*x^2 - 6*x^3 + 3*x^4) / (1 - x)^4.

%F a(n) = (8*n^3 - 15*n^2 + 15*n - 12)/2 for n>1.

%F (End)

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

%t Module[{s = 0}, (Do[If[w + x + y > 2, s = s + 1], {w, #}, {x, #}, {y, #}] &[ Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];

%t Map[t[#] &, Range[0, 60]] (* A211614 *)

%t FindLinearRecurrence[%]

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

%t Join[{0, 1},LinearRecurrence[{4, -6, 4, -1},{11, 57, 160, 344},34]] (* _Ray Chandler_, Aug 02 2015 *)

%o (PARI) concat(0, Vec(x*(1 + 7*x + 19*x^2 - 6*x^3 + 3*x^4) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 04 2017

%Y Cf. A211422.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 16 2012