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

0,2

Comments

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

For a guide to related sequences, see A211795.

Links

Table of n, a(n) for n=0..32.

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

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.: f(x)/g(x), where f(x) = 2(2*x + 10*x^2 + 31*x^3 + 40*x^4 + 36*x^5 + 18*x^6 + 7*x^7) and g(x)=((1-x)^5)((1+x)^2)(1+x+x^2).

Example

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).

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 *)

Crossrefs

Cf. A211795.

Sequence in context: A296392 A318011 A328685 * A196514 A249629 A131459

Adjacent sequences: A212897 A212898 A212899 * A212901 A212902 A212903

Keyword

nonn,easy

Author

Clark Kimberling, May 31 2012

Status

approved