A212575 Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|=|x-y|+|y-z|.

0, 1, 4, 17, 42, 85, 142, 235, 346, 495, 680, 911, 1172, 1505, 1872, 2305, 2798, 3365, 3978, 4699, 5470, 6335, 7284, 8335, 9448, 10705, 12028, 13473, 15026, 16709, 18470, 20411, 22434, 24607, 26912, 29375, 31932, 34705, 37576, 40625, 43830

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 (0, 2, 2, -1, -4, -1, 2, 2, 0, -1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 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]]   (* A212575 *)
LinearRecurrence[{0, 2, 2, -1, -4, -1, 2, 2, 0, -1}, {0, 1, 4, 17, 42, 85, 142, 235, 346, 495}, 40]

Formula

a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)-a(n-6)+2*a(n-7)+2*a(n-8)-a(n-10).

G.f.: (x + 4*x^2 + 15*x^3 + 32*x^4 + 44*x^5 + 32*x^6 + 15*x^7 + 4*x^8 + x^9)/(1 - 2*x^2 - 2*x^3 + x^4 + 4*x^5 + x^6 - 2*x^7 - 2*x^8 + x^10)