A212514 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y>3z.

0, 0, 0, 0, 14, 42, 90, 200, 364, 585, 960, 1440, 2052, 2926, 4004, 5280, 7000, 9000, 11340, 14280, 17670, 21483, 26180, 31416, 37296, 44252, 52000, 60480, 70434, 81270, 93150, 106720, 121520, 137445, 155584, 175032, 196020, 219450, 244644, 271440, 301340

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]]   (* A212514 *)
LinearRecurrence[{0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1},{0,0,0,0,14,42,90,200,364,585,960,1440,2052,2926,4004,5280},50] (* Harvey P. Dale, Dec 24 2020 *)

PARI

concat(vector(4), Vec(x^4*(14 +42*x +62*x^2 +88*x^3 +114*x^4 +103*x^5 +90*x^6 +74*x^7 +42*x^8 +15*x^9 +4*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^60))) \\ Colin Barker, Dec 18 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^4*(14 +42*x +62*x^2 +88*x^3 +114*x^4 +103*x^5 +90*x^6 +74*x^7 +42*x^8 +15*x^9 +4*x^10) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - Colin Barker, Dec 18 2015