A211624 - OEIS

%I #13 Aug 23 2017 06:22:43

%S 0,4,30,104,245,485,837,1339,1998,2858,3920,5234,6795,8659,10815,

%T 13325,16172,19424,23058,27148,31665,36689,42185,48239,54810,61990,

%U 69732,78134,87143,96863,107235,118369,130200,142844,156230,170480,185517,201469,218253

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

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

%H Colin Barker, <a href="/A211624/b211624.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

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

%F From _Colin Barker_, Aug 23 2017: (Start)

%F G.f.: x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2).

%F a(n) = (64*n^3 - 14*n^2 + 12*n) / 16 for n even.

%F a(n) = (64*n^3 - 14*n^2 + 24*n - 10) / 16 for n odd. (End)

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

%t    Module[{s = 0}, (Do[If[w + 2 x + 2 y > 0,

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

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

%t Map[t[#] &, Range[0, 60]]  (* A211624 *)

%t FindLinearRecurrence[%]

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

%t LinearRecurrence[{2, 1, -4, 1, 2, -1},{0, 4, 30, 104, 245, 485},36] (* _Ray Chandler_, Aug 02 2015 *)

%o (PARI) concat(0, Vec(x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2) + O(x^50))) \\ _Colin Barker_, Aug 23 2017

%Y Cf. A211422.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Apr 17 2012