A212960 Number of (w,x,y) with all terms in {0,...,n} and |w-x| != |x-y|.

0, 4, 16, 44, 92, 168, 276, 424, 616, 860, 1160, 1524, 1956, 2464, 3052, 3728, 4496, 5364, 6336, 7420, 8620, 9944, 11396, 12984, 14712, 16588, 18616, 20804, 23156, 25680, 28380, 31264, 34336, 37604, 41072, 44748, 48636, 52744, 57076

Offset: 0

Author

Clark Kimberling, Jun 01 2012

Keywords

• nonn
• easy

Comments

a(n)+A212959(n)=(n+1)^3. Every term is divisible by 4.

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] != Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A212960 *)
m/4 (* integers *)

Formula

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

G.f.: f(x)/g(x), where f(x)=4x(1+x^2+x^3) and g(x)=(1+x)(1-x)^4.

a(n) = (4*n^3 + 6*n^2 + 4*n+1 - (-1)^n)/4. - Luce ETIENNE, Apr 05 2014

E.g.f.: (x*(7 + 9*x + 2*x^2)*cosh(x) + (1 + 7*x + 9*x^2 + 2*x^3)*sinh(x))/2. - Stefano Spezia, Aug 11 2025