A213479 Number of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y| = w+x+y.

1, 4, 11, 18, 30, 41, 58, 73, 95, 114, 141, 164, 196, 223, 260, 291, 333, 368, 415, 454, 506, 549, 606, 653, 715, 766, 833, 888, 960, 1019, 1096, 1159, 1241, 1308, 1395, 1466, 1558, 1633, 1730, 1809, 1911, 1994, 2101, 2188, 2300, 2391, 2508, 2603, 2725, 2824

Offset: 0

Author

Clark Kimberling, Jun 13 2012

Keywords

• nonn
• easy

Comments

a(n) + A213480(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y == Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213479 *)

PARI

Vec((1+3*x+5*x^2+x^3-x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 27 2016

Formula

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

G.f.: (1 + 3*x + 5*x^2 + x^3 - x^4)/((1 - x)^3 * (1 + x)^2).

From Colin Barker, Jan 27 2016: (Start)

a(n) = (18*n^2+2*(-1)^n*n+42*n+5*(-1)^n+11)/16.

a(n) = (9*n^2+22*n+8)/8 for n even.

a(n) = (9*n^2+20*n+3)/8 for n odd. (End)