A212680 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y|=|y-z|+1.

0, 0, 4, 18, 56, 120, 228, 378, 592, 864, 1220, 1650, 2184, 2808, 3556, 4410, 5408, 6528, 7812, 9234, 10840, 12600, 14564, 16698, 19056, 21600, 24388, 27378, 30632, 34104, 37860, 41850, 46144, 50688, 55556, 60690, 66168, 71928, 78052

Offset: 0

Author

Clark Kimberling, May 24 2012

Keywords

• nonn
• easy

Comments

Every term is even. For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[x - y] == Abs[y - z] + 1, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212680 *)
%/2 (* integers *)
LinearRecurrence[{2, 1, -4, 1, 2, -1 }, {0, 0, 4, 18, 56, 120 }, 40]

Formula

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

G.f.: (2 x^2 (2 + x) (1 + 2 x + 3 x^2))/((-1 + x)^4 (1 + x)^2). [corrected by Clark Kimberling, Feb 27 2018]