A212899 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)>2.

0, 0, 0, 2, 16, 62, 186, 456, 962, 1818, 3162, 5156, 7986, 11862, 17018, 23712, 32226, 42866, 55962, 71868, 90962, 113646, 140346, 171512, 207618, 249162, 296666, 350676, 411762, 480518, 557562, 643536, 739106, 844962, 961818

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. a(n)+A212898(n)=(n+1)^4.

For a guide to related sequences, see A211795.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• 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]] > 2, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212899 *)
m/2  (* integers *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,2,16,62,186,456,962,1818},40] (* Harvey P. Dale, Jun 01 2025 *)

Formula

a(n)=5*a(n-1)+10*a(n-2)-10*a(n-3)+5*a(n-4)-a(n-5) for n>=9.

G.f.: (2*x^3 + 6*x^4 + 2*x^5 + 16*x^6 - 4*x^7 + 2*x^9 )/(1 - 5*x + 10*x^2 - 10*x^3 + 5*x^4 - x^5)