A213393 Number of (w,x,y) with all terms in {0,...,n} and 2*max(w,x,y) > 3*min(w,x,y).

0, 6, 24, 54, 108, 192, 300, 450, 648, 882, 1176, 1536, 1944, 2430, 3000, 3630, 4356, 5184, 6084, 7098, 8232, 9450, 10800, 12288, 13872, 15606, 17496, 19494, 21660, 24000, 26460, 29106, 31944, 34914, 38088, 41472, 45000, 48750, 52728

Offset: 0

Author

Clark Kimberling, Jun 11 2012

Keywords

• nonn
• easy

Comments

Every term is even.

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A212959, A213391.

Cf. A092076, A190798, A213392.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2*Max[w, x, y] > 3*Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213393 *)
m/2   (* integers *)

Formula

a(n) + A213391(n+1) = (n+1)^3.

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-4*a(n-4)+2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8).

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

From Ayoub Saber Rguez, Feb 01 2022: (Start)

a(n) = 6*A190798(n+1).

a(n) = A213392(n) - A092076(n).

a(n) = (8*n^2+16*n+8-8*n*((2*n+2) mod 3)-8*((2*n+2) mod 3)+2*((2*n+2) mod 3)^2)/3. (End)

E.g.f.: 2*exp(-x/2)*(6*exp(3*x/2)*(1 + x*(13 + 2*x*(6 + x))) - 3*(2 + x)*cos(sqrt(3)*x/2) - sqrt(3)*(2 - 3*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Feb 25 2023