A212685 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=w+|y-z|.

0, 0, 2, 7, 18, 36, 64, 103, 156, 224, 310, 415, 542, 692, 868, 1071, 1304, 1568, 1866, 2199, 2570, 2980, 3432, 3927, 4468, 5056, 5694, 6383, 7126, 7924, 8780, 9695, 10672, 11712, 12818, 13991, 15234, 16548, 17936, 19399, 20940, 22560

Offset: 0

Author

Clark Kimberling, May 24 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• 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[Abs[w - x] == w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212685 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 18}, 40]

Formula

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

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

a(n)=(-3+3*(-1)^n+4*n-6*n^2+8*n^3)/24. [Colin Barker, Jun 10 2012]