A212069 - OEIS

%I #17 May 24 2017 04:23:49

%S 0,1,2,9,22,41,72,115,170,243,334,443,576,733,914,1125,1366,1637,1944,

%T 2287,2666,3087,3550,4055,4608,5209,5858,6561,7318,8129,9000,9931,

%U 10922,11979,13102,14291,15552,16885,18290,19773,21334,22973

%N Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.

%C w is the average of {x,y,z}, as well as {w,x,y,z}.

%C For a guide to related sequences, see A211795.

%C a(n) is also the number of (w,x,y,z) with all terms in {0,1,...,n-1} and 3*w = x+y+z. - _Clark Kimberling_, May 16 2012

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

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

%F From _R. J. Mathar_, Jun 25 2012: (Start)

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

%F 3*a(n) = n^3 + 2*A049347(n-1). (End)

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

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

%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

%t Map[t[#] &, Range[0, 50]] (* A212087 *)

%t FindLinearRecurrence[%]

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

%t LinearRecurrence[{3, -3, 2, -3, 3, -1},{0, 1, 2, 9, 22, 41},42] (* _Ray Chandler_, Aug 02 2015 *)

%Y Cf. A211795.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 01 2012