A212984 - OEIS

%I #14 Sep 08 2022 08:46:02

%S 1,1,3,6,8,12,17,21,27,34,40,48,57,65,75,86,96,108,121,133,147,162,

%T 176,192,209,225,243,262,280,300,321,341,363,386,408,432,457,481,507,

%U 534,560,588,617,645,675,706,736,768,801,833,867,902,936,972,1009

%N Number of (w,x,y) with all terms in {0..n} and 3w = x+y.

%C For a guide to related sequences, see A212959.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

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

%F G.f.: f(x)/g(x), where f(x) = 1 - x + 2*x^2 and g(x) = (1+x+x^2)*(1-x)^3.

%F a(n) = 1 + floor(2*n/3) + floor(n^2/3). - _Wesley Ivan Hurt_, Jul 25 2016

%p A212984:=n->1 + floor(2*n/3) + floor(n^2/3): seq(A212984(n), n=0..100); # _Wesley Ivan Hurt_, Jul 25 2016

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[3 w == x + y, s = s + 1],

%t {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];

%t m = Map[t[#] &, Range[0, 70]]   (* A212984 *)

%o (Magma) [1 + Floor(2*n/3) + Floor(n^2/3) : n in [0..80]]; // _Wesley Ivan Hurt_, Jul 25 2016

%Y Cf. A212959.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Jun 04 2012