A212896 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)<2.

1, 16, 79, 240, 551, 1066, 1839, 2924, 4375, 6246, 8591, 11464, 14919, 19010, 23791, 29316, 35639, 42814, 50895, 59936, 69991, 81114, 93359, 106780, 121431, 137366, 154639, 173304, 193415, 215026, 238191, 262964, 289399, 317550

Offset: 0

Author

Clark Kimberling, May 31 2012

Keywords

• nonn
• easy

Comments

The gapsizes are |w-x|, |x-y|, |y-z|.

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:=[1, 16, 79, 240, 551, 1066]; [n le 6 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]] <= 1, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212896 *)
CoefficientList[Series[(1+12*x+21*x^2+16*x^3+2*x^4+2*x^5) /(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>=6.

G.f.: f(x)/g(x), where f(x) = 1+12*x+21*x^2+16*x^3+2*x^4+2*x^5 and g(x) = (1-x)^4.

a(n) = 9*n^3-6*n^2+20*n-9 with n>1, a(0)=1, a(1)=16. - Bruno Berselli, Jun 12 2012