A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3*w = x+y+z.

0, 1, 2, 9, 22, 41, 72, 115, 170, 243, 334, 443, 576, 733, 914, 1125, 1366, 1637, 1944, 2287, 2666, 3087, 3550, 4055, 4608, 5209, 5858, 6561, 7318, 8129, 9000, 9931, 10922, 11979, 13102, 14291, 15552, 16885, 18290, 19773, 21334, 22973

Offset: 0

Author

Clark Kimberling, May 01 2012

Keywords

• nonn
• easy

Comments

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

For a guide to related sequences, see A211795.

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

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A049347, A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212087 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 2, -3, 3, -1},{0, 1, 2, 9, 22, 41},42] (* Ray Chandler, Aug 02 2015 *)

Formula

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).

From R. J. Mathar, Jun 25 2012: (Start)

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

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

E.g.f.: (3*exp(x)*x*(1 + 3*x + x^2) + 4*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Aug 06 2025