A212516 Number of (w,x,y,z) with all terms in {1,...,n} and w>2x and y<3z.
0, 0, 0, 8, 28, 88, 186, 378, 660, 1104, 1700, 2575, 3660, 5148, 6972, 9310, 12096, 15616, 19656, 24624, 30330, 37100, 44770, 53845, 63888, 75600, 88608, 103428, 119756, 138376, 158550, 181350, 206160, 233728, 263568, 296803, 332316, 371628, 413820, 459914
Offset: 0
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).
Programs
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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]] (* A212516 *) Table[-n^2/24 - n^3/6 + 5*n^4/24 - 1/12*Floor[n/6] + (n/6 - n^2/12)*Floor[n/3] - (n/12 + 5*n^2/12)*Floor[n/2] + 1/12*Floor[(1 + n)/6] + (-n/6 + n^2/12) * Floor[(1 + n)/3], {n, 0, 50}] (* Vaclav Kotesovec, Dec 11 2015 *) -
PARI
concat(vector(3), Vec(x^3*(8 +28*x +72*x^2 +114*x^3 +154*x^4 +172*x^5 +176*x^6 +146*x^7 +113*x^8 +64*x^9 +26*x^10 +6*x^11 +x^12) / ((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*(8 +28*x +72*x^2 +114*x^3 +154*x^4 +172*x^5 +176*x^6 +146*x^7 +113*x^8 +64*x^9 +26*x^10 +6*x^11 +x^12) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - Colin Barker, Dec 11 2015
Comments