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.

1, 2, 10, 48, 208, 874, 3600, 14622, 58976, 236824, 949094, 3800126, 15207812, 60846154, 243414326, 973716670, 3894985588, 15580180122, 62321195992, 249285735518, 997144844044, 3988583179554, 15954340324098, 63817376508852, 255269536476262

Offset

0,2

Comments

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.

See illustration in links.

From Jon E. Schoenfield, Mar 23 2014: (Start)

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).

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)

After Wolfdieter Lang's suggestions of an alternative definition, I found later that a(n) = A053417(2^n-1), n >= 1.

Links

Table of n, a(n) for n=0..24.

Kival Ngaokrajang, Illustration of initial terms

Kival Ngaokrajang, Small Basic program

Formula

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

Prog

(Small Basic)
See links.
(Magma)
for n in [0..25] do
   x:=(2^n - 1)^2;
   c:=2^n; // for row i=0
   for i in [1..Isqrt(x div 3)] do
      t:=x-3*i*i;
      if IsOdd(i) then
         c:=c+2*(2*Floor(Sqrt(t)/2)+1);
      else
         c:=c+2*(2*Floor(Sqrt(t)/2+1/2));
      end if;
   end for;
   n, c;
end for; // Jon E. Schoenfield, Mar 17 2014

Crossrefs

Cf. A239074, A239206, A053417, A053416.

Sequence in context: A054138 A192696 A181296 * A065982 A114693 A086853

Adjacent sequences: A239070 A239071 A239072 * A239074 A239075 A239076

Keyword

nonn

Author

Kival Ngaokrajang, Mar 10 2014

Extensions

a(20) - a(24) and corrections to a(15), a(17), and a(18) from Jon E. Schoenfield, Mar 17 2014

Status

approved