A212574 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|>=|x-y|+|y-z|.

0, 1, 8, 33, 88, 197, 380, 673, 1104, 1721, 2560, 3681, 5128, 6973, 9268, 12097, 15520, 19633, 24504, 30241, 36920, 44661, 53548, 63713, 75248, 88297, 102960, 119393, 137704, 158061, 180580, 205441, 232768, 262753, 295528, 331297

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• 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]]   (* A212574 *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 1, 8, 33, 88, 197, 380}, 40]

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).

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