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A239073 Given a circle of radius R into which small circles of radius R/2^n are packed in a "hexagonal pattern" (see Comments), a(n) is the maximum number of small circles.

%I A239073 #33 Sep 08 2022 08:46:07
%S A239073 1,2,10,48,208,874,3600,14622,58976,236824,949094,3800126,15207812,
%T A239073 60846154,243414326,973716670,3894985588,15580180122,62321195992,
%U A239073 249285735518,997144844044,3988583179554,15954340324098,63817376508852,255269536476262
%N A239073 Given a circle of radius R into which small circles of radius R/2^n are packed in a "hexagonal pattern" (see Comments), a(n) is the maximum number of small circles.
%C A239073 The construction rule is: (1) Start with a unit circle (big circle). (2) Pack circles at radius 1/2^n (small circles) on the diameter line of the big circle. (3) Maintain hexagonal packing pattern of small circles on the rows above and below that of the previous step. The number of small circles in any row is limited so that the circumference of the last small circle does not cross the circumference of the big circle (but is allowed to contact it). (4) Repeat process to the top and bottom rows.
%C A239073 See illustration in links.
%C A239073 From _Jon E. Schoenfield_, Mar 23 2014: (Start)
%C A239073 There can never be exactly one small circle on the top row iMax (and, symmetrically, exactly one on the bottom row) that contacts the large circle, since the upper edge of that small circle would be at a radius of (iMax*sqrt(3)+1)/2^n from the center of the unit circle (which cannot equal 1 unless iMax=0, i.e., there are no rows above or below the center).
%C A239073 The number of small circles on the top row seems to be on the order of 2^(n/2). It seems nearly certain that, for n>0, there is never exactly one small circle on the top row. (End)
%C A239073 After _Wolfdieter Lang_'s suggestions of an alternative definition, I found later that a(n) = A053417(2^n-1), n >= 1.
%H A239073 Kival Ngaokrajang, <a href="/A239073/a239073.pdf">Illustration of initial terms</a>
%H A239073 Kival Ngaokrajang, <a href="/A239073/a239073.txt">Small Basic program</a>
%F A239073 a(n) = 2^n + 2 * Sum_{i odd, <= iMax} (2*floor(sqrt(x - 3 * i^2) / 2) + 1) + 2 * Sum_{i even, 0 < i <= iMax} 2*floor(sqrt(x - 3 * i^2) / 2 + 1/2) where x = (2^n-1)^2 and iMax = floor(sqrt(x/3)). - _Jon E. Schoenfield_, Mar 17, Mar 23 2014
%o A239073 (Small Basic)
%o A239073 See links.
%o A239073 (Magma)
%o A239073 for n in [0..25] do
%o A239073    x:=(2^n - 1)^2;
%o A239073    c:=2^n; // for row i=0
%o A239073    for i in [1..Isqrt(x div 3)] do
%o A239073       t:=x-3*i*i;
%o A239073       if IsOdd(i) then
%o A239073          c:=c+2*(2*Floor(Sqrt(t)/2)+1);
%o A239073       else
%o A239073          c:=c+2*(2*Floor(Sqrt(t)/2+1/2));
%o A239073       end if;
%o A239073    end for;
%o A239073    n, c;
%o A239073 end for; // _Jon E. Schoenfield_, Mar 17 2014
%Y A239073 Cf. A239074, A239206, A053417, A053416.
%K A239073 nonn
%O A239073 0,2
%A A239073 _Kival Ngaokrajang_, Mar 10 2014
%E A239073 a(20) - a(24) and corrections to a(15), a(17), and a(18) from _Jon E. Schoenfield_, Mar 17 2014