A212573 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|>|x-y|+|y-z|.

0, 0, 2, 10, 36, 92, 202, 386, 680, 1112, 1730, 2570, 3692, 5140, 6986, 9282, 12112, 15536, 19650, 24522, 30260, 36940, 44682, 53570, 63736, 75272, 88322, 102986, 119420, 137732, 158090, 180610, 205472, 232800, 262786, 295562, 331332

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Bo Gyu Jeong, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A211795, A212570.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] > Abs[x - y] + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212573 *)
%/2  (* integers *)

Formula

a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).

From Bruno Berselli, Jun 14 2012: (Start)

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

a(n) = (2*(n-3)*n*(5*n^2-15*n+31)-3*(2*n-3)*(-1)^n+87)/48. (End)