A213484 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| + |y-w| >= w+x+y.

1, 4, 7, 10, 16, 25, 34, 43, 55, 70, 85, 100, 118, 139, 160, 181, 205, 232, 259, 286, 316, 349, 382, 415, 451, 490, 529, 568, 610, 655, 700, 745, 793, 844, 895, 946, 1000, 1057, 1114, 1171, 1231, 1294, 1357, 1420, 1486, 1555, 1624, 1693, 1765

Offset: 0

Author

Clark Kimberling, Jun 13 2012

Keywords

• nonn
• easy

Comments

a(n) + A213485(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y == Abs[w - x] + Abs[x - y] + Abs[y - w],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213484 *)

Formula

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

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

From Ayoub Saber Rguez, Dec 31 2021: (Start)

a(n) + A213485(n) = (n+1)^3.

a(n) = 3*A054925(n+1) + 1.

a(n) = 3*(A192447(n+1)/2) + 1.

a(n) = (3*n^2 + 3*n + 4 + 3*((n+1) mod 4 - (n+1) mod 2))/4. (End)