A212900 Number of (w,x,y,z) with all terms in {0,...,n} and distinct consecutive gap sizes.

0, 4, 28, 122, 340, 786, 1558, 2814, 4690, 7404, 11130, 16140, 22652, 30992, 41416, 54310, 69968, 88830, 111234, 137674, 168526, 204344, 245542, 292728, 346360, 407100, 475444, 552114, 637644, 732810, 838190, 954614, 1082698

Offset: 0

Author

Clark Kimberling, May 31 2012

Keywords

• nonn
• easy

Comments

The gap sizes are |w-x|, |x-y|, |y-z|. Every term is even.

For a guide to related sequences, see A211795.

Examples

			a(1)=4 counts these (w,x,y,z): (0,0,1,1), (0,1,1,0), (1,1,0,0), (1,0,0,1).
		

Links

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

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] != Abs[x - y] && Abs[x - y] != Abs[y - z], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212900 *)
m/2  (* integers *)
LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{0,4,28,122,340,786,1558,2814,4690},40] (* Harvey P. Dale, Aug 25 2013 *)

Formula

a(n) = 2*a(n-1)+a(n-2)-3*a(n-3)-a(n-4)+a(n-5)+3*a(n-6)-a(n-7)-2*a(n-8)+a(n-9).

G.f.: 2*x*(2 + 10*x + 31*x^2 + 40*x^3 + 36*x^4 + 18*x^5 + 7*x^6)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)).