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A212505 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>=2z.

%I A212505 #15 Mar 09 2016 05:24:55
%S A212505 0,0,3,14,48,114,243,444,768,1220,1875,2730,3888,5334,7203,9464,12288,
%T A212505 15624,19683,24390,30000,36410,43923,52404,62208,73164,85683,99554,
%U A212505 115248,132510,151875,173040,196608,222224,250563,281214,314928
%N A212505 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>=2z.
%C A212505 For a guide to related sequences, see A211795.
%H A212505 Alois P. Heinz, <a href="/A212505/b212505.txt">Table of n, a(n) for n = 0..1000</a>
%H A212505 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A212505 a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).
%F A212505 From _Alois P. Heinz_, May 31 2012: (Start)
%F A212505 a(n) = A077043(n) * A002620(n) = floor(n^2/4)*ceiling(n^2*3/4).
%F A212505 G.f.: x^2*(x^2+2*x+3)*(3*x^2+2*x+1) / ((x+1)^3*(1-x)^5). (End)
%p A212505 a:= n-> floor(n^2/4)*ceil(n^2*3/4):
%p A212505 seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 13 2013
%t A212505 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212505 (Do[If[w < 2 x && y >= 2 z, s = s + 1],
%t A212505 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212505 Map[t[#] &, Range[0, 40]]   (* A212505 *)
%Y A212505 Cf. A211795.
%K A212505 nonn,easy
%O A212505 0,3
%A A212505 _Clark Kimberling_, May 19 2012