A212572 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| <= |x-y| + |y-z|.

0, 1, 14, 71, 220, 533, 1094, 2015, 3416, 5449, 8270, 12071, 17044, 23421, 31430, 41343, 53424, 67985, 85326, 105799, 129740, 157541, 189574, 226271, 268040, 315353, 368654, 428455, 495236, 569549, 651910, 742911, 843104, 953121, 1073550, 1205063, 1348284

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Colin Barker, 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]]   (* A212572 *)

PARI

concat(0, Vec(x*(1+11*x+30*x^2+26*x^3+9*x^4-x^5)/((1-x)^5*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 06 2015

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 Colin Barker, Dec 06 2015: (Start)

a(n) = 1/48*(38*n^4+20*n^3-32*n^2-2*(3*(-1)^n-11)*n+3*((-1)^n-1)).

G.f.: x*(1+11*x+30*x^2+26*x^3+9*x^4-x^5) / ((1-x)^5*(1+x)^2).

(End)