A212897 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)>1.

0, 0, 2, 16, 74, 230, 562, 1172, 2186, 3754, 6050, 9272, 13642, 19406, 26834, 36220, 47882, 62162, 79426, 100064, 124490, 153142, 186482, 224996, 269194, 319610, 376802, 441352, 513866, 594974, 685330, 785612, 896522, 1018786

Offset: 0

Author

Clark Kimberling, May 31 2012

Keywords

• nonn
• easy

Comments

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

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] > 1, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212897 *)
m/2 (* integers *)

Formula

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>=7.

G.f.: (2*x^2 + 6*x^3 + 14*x^4 + 2*x^6)/(1 - 5*x + 10*x^2 - 10*x^3 + 5*x^4 - x^5).

a(n) = n^4-5*n^3+12*n^2-16*n+10 with n>1, a(0)=a(1)=0. [Bruno Berselli, Jun 12 2012]