A212513 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y<=3z.

0, 1, 16, 72, 210, 483, 990, 1760, 2964, 4680, 7040, 10176, 14364, 19551, 26180, 34320, 44200, 56025, 70308, 86800, 106330, 128898, 154836, 184416, 218448, 256373, 299520, 347760, 401534, 461175, 527850, 600576, 681296, 769692, 866320, 971568, 1087020

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]]   (* A212513 *)

PARI

concat(0, Vec(x*(1 +16*x +70*x^2 +176*x^3 +308*x^4 +446*x^5 +510*x^6 +514*x^7 +471*x^8 +372*x^9 +220*x^10 +102*x^11 +30*x^12 +4*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^100))) \\ 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*(1 +16*x +70*x^2 +176*x^3 +308*x^4 +446*x^5 +510*x^6 +514*x^7 +471*x^8 +372*x^9 +220*x^10 +102*x^11 +30*x^12 +4*x^13) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - Colin Barker, Dec 18 2015