A212989 Number of (w,x,y) with all terms in {0,...,n} and 4*w = 4*x+y.

1, 2, 3, 4, 9, 11, 13, 15, 24, 27, 30, 33, 46, 50, 54, 58, 75, 80, 85, 90, 111, 117, 123, 129, 154, 161, 168, 175, 204, 212, 220, 228, 261, 270, 279, 288, 325, 335, 345, 355, 396, 407, 418, 429, 474, 486, 498, 510, 559, 572, 585, 598, 651, 665, 679, 693

Offset: 0

Author

Clark Kimberling, Jun 04 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2, 0, 0, -1, 1).

Crossrefs

Cf. A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[4 w == 4 x + y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212989 *)
LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{1,2,3,4,9,11,13,15,24},60] (* Harvey P. Dale, Sep 20 2023 *)

Formula

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

G.f.: f(x)/g(x), where f(x) = 1 + x + x^2 + x^3 + 3*x^4 and g(x) = ((1 + x + x^2 + x^3)^2)(1-x)^3.