A213488 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| + |y-w| < w+x+y.

0, 1, 8, 27, 61, 113, 189, 295, 434, 609, 826, 1091, 1407, 1777, 2207, 2703, 3268, 3905, 4620, 5419, 6305, 7281, 8353, 9527, 10806, 12193, 13694, 15315, 17059, 18929, 20931, 23071, 25352, 27777, 30352, 33083, 35973, 39025, 42245, 45639

Offset: 0

Author

Clark Kimberling, Jun 13 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

Crossrefs

Cf. A212959, A213489.

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)]];
m = Map[t[#] &, Range[0, 60]]
LinearRecurrence[{4,-7,8,-7,4,-1},{0,1,8,27,61,113},40] (* Harvey P. Dale, Sep 10 2019 *)

Formula

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

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

a(n) = (n+1)^3 - A213489(n).