A212523 Number of (w,x,y,z) with all terms in {1,...,n} and w+x

0, 0, 5, 31, 106, 270, 575, 1085, 1876, 3036, 4665, 6875, 9790, 13546, 18291, 24185, 31400, 40120, 50541, 62871, 77330, 94150, 113575, 135861, 161276, 190100, 222625, 259155, 300006, 345506, 395995, 451825, 513360, 580976, 655061

Offset: 0

Author

Clark Kimberling, May 21 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

If the initial 0 is omitted, the sequence {b(n): n>=0} = {0, 5, 31, 106, 270, 575, 1085, 1876, 3036, 4665, 6875, 9790, ...} is given by b(n) = n*(n+1)*(3*n^2+7*n+5)/6. - N. J. A. Sloane, Jul 25 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A211795.

Programs

Magma

I:=[0, 0, 5, 31, 106]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2) +10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jun 09 2012

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x < y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212523 *)
LinearRecurrence[{5,-10, 10, -5, 1},{0,0,5,31,106},60] (* Vincenzo Librandi, Jun 09 2012 *)

Formula

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

a(n) = Sum_{i=0..n-1} A048395(i). - J. M. Bergot, Jun 08 2012

G.f.: -x^2*(x+5)*(1+x)/(x-1)^5 . - R. J. Mathar, Sep 23 2016