A213397 Number of (w,x,y) with all terms in {0,...,n} and 2*w >= |x+y-z|.

1, 5, 18, 43, 83, 144, 229, 341, 486, 667, 887, 1152, 1465, 1829, 2250, 2731, 3275, 3888, 4573, 5333, 6174, 7099, 8111, 9216, 10417, 11717, 13122, 14635, 16259, 18000, 19861, 21845, 23958, 26203, 28583, 31104, 33769, 36581, 39546, 42667, 45947, 49392, 53005, 56789, 60750

Offset: 0

Author

Clark Kimberling, Jun 12 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A212959, A213396.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w >= Abs[x + y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212297 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 - x)^4*(1 + x + x^2)), {x, 0, 44}], x] (* Michael De Vlieger, Dec 22 2017 *)

PARI

first(n) = Vec((1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x + x^2)) + O(x^n)) \\ Iain Fox, Dec 22 2017

Formula

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

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

a(n) = (n+1)^3 - A213396(n).

a(n) = floor(2*n^3/3) + 2*n*(n + 1) + 1. - Bruno Berselli, Dec 22 2017