A212562 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<2y+2z.

0, 1, 15, 73, 228, 551, 1137, 2097, 3568, 5701, 8675, 12681, 17940, 24683, 33173, 43681, 56512, 71977, 90423, 112201, 137700, 167311, 201465, 240593, 285168, 335661, 392587, 456457, 527828, 607251, 695325, 792641, 899840, 1017553, 1146463, 1287241, 1440612

Offset: 0

Author

Clark Kimberling, May 21 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A211795.

Programs

Magma

I:=[0,1,15,73,228,551,1137]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)-5*Self(n-3)+5*Self(n-4)+Self(n-5)-3*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 05 2015

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x < 2 y + 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212562 *)
CoefficientList[Series[x (1 + 12 x + 29 x^2 + 29 x^3 + 10 x^4 + x^5)/((1 - x)^5 (1 + x)^2), {x, 0, 33}], x] (* Vincenzo Librandi, Dec 05 2015 *)

PARI

concat(0, Vec(x*(1+12*x+29*x^2+29*x^3+10*x^4+x^5)/((1-x)^5*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 05 2015

Formula

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).

From Colin Barker, Dec 05 2015: (Start)

a(n) = 1/96*(82*n^4 + 12*n^3 + 8*n^2 + 6*((-1)^n-1)*n - 3*(-1)^n + 3).

G.f.: x*(1 + 12*x + 29*x^2 + 29*x^3 + 10*x^4 + x^5) / ((1-x)^5*(1+x)^2). (End)