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A212506 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y<=2z.

%I A212506 #14 Mar 09 2016 05:23:42
%S A212506 0,1,16,64,196,441,900,1600,2704,4225,6400,9216,12996,17689,23716,
%T A212506 30976,40000,50625,63504,78400,96100,116281,139876,166464,197136,
%U A212506 231361,270400,313600,362404,416025,476100,541696,614656,693889,781456
%N A212506 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y<=2z.
%C A212506 For a guide to related sequences, see A211795.
%H A212506 <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 A212506 a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).
%F A212506 From _Alois P. Heinz_, May 31 2012: (Start)
%F A212506 a(n) = A006578(n)^2.
%F A212506 G.f.: x*(14*x+30*x^2+42*x^3+17*x^4+4*x^5+1) / ((x+1)^3*(1-x)^5). (End)
%p A212506 a:= n-> (n*(n+1)/2+floor(n^2/4))^2:
%p A212506 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 31 2012
%t A212506 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212506 (Do[If[w <= 2 x && y <= 2 z, s = s + 1],
%t A212506 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212506 Map[t[#] &, Range[0, 40]]   (* A212506 *)
%Y A212506 Cf. A211795.
%K A212506 nonn,easy
%O A212506 0,3
%A A212506 _Clark Kimberling_, May 19 2012