A212570 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=|x-y|+|y-z|.

0, 1, 6, 23, 52, 105, 178, 287, 424, 609, 830, 1111, 1436, 1833, 2282, 2815, 3408, 4097, 4854, 5719, 6660, 7721, 8866, 10143, 11512, 13025, 14638, 16407, 18284, 20329, 22490, 24831, 27296, 29953, 32742, 35735, 38868, 42217, 45714, 49439

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Apart from the first term, partial sums of A220082. [Bruno Berselli, Dec 05 2012]

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A211795.

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]]   (* A212570 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,1,6,23,52,105},40] (* Harvey P. Dale, Oct 02 2021 *)

Formula

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

a(n) = n*(-1-3*(-1)^n+10*n^2)/12. G.f.: x*(x^4+4*x^3+10*x^2+4*x+1)/((x-1)^4*(x+1)^2). [Colin Barker, Oct 04 2012]