A212675 Number of (w,x,y,z) with all terms in {1,...,n} and w >= |x-y| + |y-z|.

0, 1, 14, 57, 158, 353, 688, 1217, 2004, 3121, 4650, 6681, 9314, 12657, 16828, 21953, 28168, 35617, 44454, 54841, 66950, 80961, 97064, 115457, 136348, 159953, 186498, 216217, 249354, 286161, 326900, 371841, 421264, 475457, 534718

Offset: 0

Author

Clark Kimberling, May 23 2012

Keywords

• nonn
• easy

Comments

a(n) + A212568(n) = n^4.

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (4, -5, 0, 5, -4, 1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w >= Abs[x - y] + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212675 *)
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 14, 57, 158, 353}, 40]

Formula

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

G.f.: (x + 10*x^2 + 6*x^3 + x^5)/(1 - 4*x + 5*x^2 - 5*x^4 + 4*x^5 - x^6). [corrected by Georg Fischer, May 10 2019]