A212068 Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.

0, 0, 3, 10, 25, 49, 86, 137, 206, 294, 405, 540, 703, 895, 1120, 1379, 1676, 2012, 2391, 2814, 3285, 3805, 4378, 5005, 5690, 6434, 7241, 8112, 9051, 10059, 11140, 12295, 13528, 14840, 16235, 17714, 19281, 20937, 22686, 24529, 26470, 28510, 30653, 32900

Offset: 0

Author

Clark Kimberling, May 01 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w == x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212068 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -2, -2, 3, -1},{0, 0, 3, 10, 25},42] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(vector(2), Vec(x^2*(3 + x + x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

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

From Colin Barker, Dec 02 2017: (Start)

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

a(n) = n*(10*n^2 - 3*n + 2)/24 for n even.

a(n) = (n - 1)*(10*n^2 + 7*n + 9)/24 for n odd.

(End)