A212965 Number of triples (w,x,y) with all terms in {0,...,n} and such that w = max(w,x,y) - min(w,x,y).

1, 4, 12, 21, 37, 52, 76, 97, 129, 156, 196, 229, 277, 316, 372, 417, 481, 532, 604, 661, 741, 804, 892, 961, 1057, 1132, 1236, 1317, 1429, 1516, 1636, 1729, 1857, 1956, 2092, 2197, 2341, 2452, 2604, 2721, 2881, 3004, 3172, 3301, 3477, 3612

Offset: 0

Author

Clark Kimberling, Jun 02 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Links

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

Crossrefs

Cf. A212959, A213498.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w == Max[w, x, y] - Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]   (* A212965 *)

Formula

a(n) = (14*n*(n+1) + (2*n+1)*(-1)^n + 7)/8.

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

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

From Ayoub Saber Rguez, Dec 06 2021: (Start)

a(n) + A213498(n) = (n+1)^3.

a(n) = (7*n^2 + 8*n + 4 - (2*n+1)*(n mod 2))/4. (End)