A212895 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)=2.

0, 0, 2, 14, 58, 168, 376, 716, 1224, 1936, 2888, 4116, 5656, 7544, 9816, 12508, 15656, 19296, 23464, 28196, 33528, 39496, 46136, 53484, 61576, 70448, 80136, 90676, 102104, 114456, 127768, 142076, 157416, 173824, 191336, 209988

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A211795.

Programs

Magma

I:=[0, 0, 2, 14, 58, 168, 376, 716, 1224]; [n le 9 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

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]]   (* A212895 *)
m/2 (* integers *)
CoefficientList[Series[2 x^2*(1+3 x+7 x^2+6 x^3-x^4+x^5+x^6) /(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

Formula

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=9.

G.f.: f(x)/g(x), where f(x)=(2 x^2)*(1 + 3 x + 7 x^2 + 6 x^3 - x^4 + x^5 + x^6) and g(x)=(1 - x)^4.