A212714 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|.

0, 0, 2, 10, 32, 78, 162, 300, 512, 820, 1250, 1830, 2592, 3570, 4802, 6328, 8192, 10440, 13122, 16290, 20000, 24310, 29282, 34980, 41472, 48828, 57122, 66430, 76832, 88410, 101250, 115440, 131072, 148240, 167042, 187578, 209952

Offset: 0

Author

Clark Kimberling, May 24 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and two squares have one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A211795, A054772.

Programs

Magma

I:=[0,0,2,10,32,78]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+5*Self(n-4)-4*Self(n-5)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Aug 02 2013

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] >= w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212714 *)
%/2  (* A011864 except for offset *)
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 0, 2, 10, 32, 78}, 40]
CoefficientList[Series[(2 x^2 + 2 x^3 + 2 x^4) / (1 - 4 x + 5 x^2 - 5 x^4 + 4 x^5 - x^6), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 02 2013 *)

Formula

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

G.f.: (2*x^2 + 2*x^3 + 2*x^4)/(1 - 4*x + 5*x^2 - 5*x^4 + 4*x^5 - x^6).

a(n) = floor(n^4/8). - Wesley Ivan Hurt, Jul 14 2013

a(n) = A054772(n, 2) = A054772(n, n^2-2), n >= 2. - Wolfdieter Lang, Oct 03 2016