A212681 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y|<|y-z|.

0, 0, 4, 24, 88, 230, 504, 966, 1696, 2772, 4300, 6380, 9144, 12714, 17248, 22890, 29824, 38216, 48276, 60192, 74200, 90510, 109384, 131054, 155808, 183900, 215644, 251316, 291256, 335762, 385200, 439890, 500224, 566544, 639268

Offset: 0

Author

Clark Kimberling, May 24 2012

Keywords

• nonn
• easy

Comments

Also, the number of (w,x,y,z) with all terms in {1,...,n} and |x-y|>|y-z|.

Every term is even.

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, A212682.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[x - y] < Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212681 *)
%/2 (* integers *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 0, 4, 24, 88, 230, 504}, 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.: (4*x^2 + 12*x^3 + 20*x^4 + 10*x^5 + 2*x^6)/(1 - 3*x + x^2 + 5*x^3 - 5*x^4 - x^5 + 3*x^6 - x^7).

a(n) + A212682(n) = n^4.