A212135 Number of (w,x,y,z) with all terms in {1,...,n} and median

0, 0, 4, 24, 84, 220, 480, 924, 1624, 2664, 4140, 6160, 8844, 12324, 16744, 22260, 29040, 37264, 47124, 58824, 72580, 88620, 107184, 128524, 152904, 180600, 211900, 247104, 286524, 330484, 379320, 433380, 493024, 558624, 630564, 709240, 795060, 888444

Offset: 0

Author

Clark Kimberling, May 05 2012

Keywords

• nonn
• easy

Comments

Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>mean.

For a guide to related sequences, see A211795.

Also, a(n+1) is the length of the long leg of the unique primitive Pythagorean triple whose inradius is A000217(n). - Miguel-Ángel Pérez García-Ortega, Jul 13 2025

Links

• Colin Barker, Table of n, a(n) for n = 0..1001
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[
Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 > (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 20]]]  (* A212135 *)
%/4 (* A002817 *)

PARI

concat(vector(2), Vec(4*x*(1 + x + x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n) + A212134(n) = n^4.

a(n) = n*(n - 1)*(n^2 - n + 2)/2.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: 4*x*(1 + x + x^2) / (1 - x)^5. - Colin Barker, Dec 02 2017