A211623 - OEIS

%I #11 Dec 05 2017 08:19:33

%S 0,2,12,28,54,86,128,176,234,298,372,452,542,638,744,856,978,1106,

%T 1244,1388,1542,1702,1872,2048,2234,2426,2628,2836,3054,3278,3512,

%U 3752,4002,4258,4524,4796,5078,5366,5664,5968,6282,6602,6932,7268,7614,7966,8328

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

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

%H Colin Barker, <a href="/A211623/b211623.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 (2,0,-2,1).

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

%F From _Colin Barker_, Dec 05 2017: (Start)

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

%F a(n) = 4*n^2 - 3*n + 2 for n>0 and even.

%F a(n) = 4*n^2 - 3*n + 1 for n odd.

%F (End)

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

%t    Module[{s = 0}, (Do[If[-1 <= w + 2 x + 3 y <= 1,

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

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

%t Map[t[#] &, Range[0, 70]]  (* A211623 *)

%t %/2  (* integers *)

%t FindLinearRecurrence[%]

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

%t Join[{0},LinearRecurrence[{2, 0, -2, 1},{2, 12, 28, 54},43]] (* _Ray Chandler_, Aug 02 2015 *)

%o (PARI) concat(0, Vec(2*x*(1 + 4*x + 2*x^2 + x^3) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ _Colin Barker_, Dec 05 2017

%Y Cf. A211422.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Apr 16 2012