A212519 Number of (w,x,y,z) with all terms in {1,...,n} and w>2x and y>=3z.

0, 0, 0, 1, 4, 12, 30, 63, 108, 192, 300, 450, 660, 936, 1260, 1715, 2240, 2880, 3672, 4617, 5670, 7000, 8470, 10164, 12144, 14400, 16848, 19773, 22932, 26460, 30450, 34875, 39600, 45056, 50864, 57222, 64260, 71928, 80028, 89167, 98800, 109200, 120540

Offset: 0

Author

Clark Kimberling, May 20 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 (0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1).

Crossrefs

Cf. A211795, A212508.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w > 2 x && y >= 3 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]]   (* A212519 *)

PARI

concat(vector(3), Vec(x^3*(1 +4*x +10*x^2 +20*x^3 +32*x^4 +32*x^5 +34*x^6 +34*x^7 +25*x^8 +14*x^9 +8*x^10 +2*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^100))) \\ Colin Barker, Dec 11 2015

Formula

a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)+2*a(n-7)-2*a(n-9)+4*a(n-11)+ a(n-12)-2*a(n-13)-2*a(n-14)+a(n-16).

G.f.: x^3*(1 +4*x +10*x^2 +20*x^3 +32*x^4 +32*x^5 +34*x^6 +34*x^7 +25*x^8 +14*x^9 +8*x^10 +2*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - Colin Barker, Dec 11 2015