A053416 -id:A053416 - cp's OEIS frontend

A127403 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at a grid point.

1, 1, 4, 4, 13, 13, 25, 31, 40, 46, 61, 73, 85, 103, 124, 130, 163, 169, 199, 211, 244, 262, 295, 319, 343, 385, 406, 436, 481, 505, 547, 577, 622, 646, 697, 739, 775, 829, 868, 916, 979, 1015, 1075, 1111, 1174, 1204, 1285, 1333, 1387, 1453, 1510, 1558, 1639

Offset: 0

Hugo Pfoertner, Feb 08 2007

nonn

			a(2)=4 because a disk of diameter 2 covers the center of the circle and the 3 net points at distance 1.

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A127402, A127404, A127405, A127406. The corresponding sequences for the square lattice and hexagonal lattice are A053411 and A053416, respectively.

a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2 && Mod[i - j , 3] != 1], {i, -n, n}, {j, -n, n}];
Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

a(n) = sum(i=-n, n, sum(j=-n, n, 4*(i^2 + i*j + j^2) <= n^2 && (i-j) % 3 != 1)); \\ Andrew Howroyd, Sep 16 2017

a(0) and terms a(23) and beyond from Andrew Howroyd, Sep 16 2017

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.

Original entry on oeis.org

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

Kival Ngaokrajang, Mar 10 2014

nonn

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.

Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Small Basic program

Cf. A239074, A239206, A053417, A053416.

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

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

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

A308685 The number of triangular lattice points whose Euclidean distance from the origin is less than or equal to n.

Original entry on oeis.org

1, 7, 19, 37, 61, 91, 127, 187, 241, 301, 367, 439, 517, 613, 721, 823, 931, 1045, 1165, 1303, 1459, 1615, 1765, 1921, 2083, 2263, 2455, 2653, 2857, 3055, 3259, 3481, 3697, 3949, 4195, 4447, 4693, 4957, 5239, 5527, 5815, 6109, 6409, 6715, 7015, 7333, 7681

Offset: 0

Peter Kagey, Jun 28 2019

nonn

Also the sum of the first n^2+1 terms of A004016.

Peter Kagey, Table of n, a(n) for n = 0..1000
Code Golf Stack Exchange, Triangular Lattice Points close to the Origin

Cf. A004016, A053416.

a(n)=1+2*vecsum(Vec(qfrep([2,1;1,2],n^2,1))) /* Georg Fischer, Apr 07 2022 (from codegolf link) */

a(n) = A053416(2*n).

A240501 Given circular disks of radius r in a hexagonal lattice covered by a circular disk of radius R = r*2n, if the center of the circle is chosen at the middle between two lattice points, a(n) is the number of points at which an r-circle is tangent to the R-circle.

Original entry on oeis.org

2, 2, 2, 6, 2, 2, 6, 2, 2, 6, 6, 2, 2, 2, 2, 6, 2, 6, 6, 6, 2, 6, 2, 2, 10, 2, 2, 2, 6, 2, 6, 6, 6, 6, 2, 2, 6, 2, 6, 6, 2, 2, 2, 2, 2, 18, 6, 6, 6, 2, 2, 6, 6, 2, 6, 6, 2, 2, 6, 6, 2, 2, 2, 6, 6, 2, 18, 2, 2, 6, 2, 6, 2, 10, 2, 6, 2, 6, 6, 2, 6, 6, 2, 2, 10, 6, 2, 6, 2, 2, 6, 6, 6, 2, 6, 2, 6, 6, 2, 6, 6, 6, 2, 2, 6, 6, 2, 6, 18, 6, 6, 6, 2, 2, 6, 6, 2, 2, 6, 2, 6, 2, 10, 18, 2, 2, 2, 2, 2, 18, 2, 2, 2, 2, 2, 6, 18, 2, 6, 6, 2, 6, 6, 6

Offset: 1

Kival Ngaokrajang, Apr 06 2014

nonn

a(n) are even R that give a(n) >= 2, which seems to be nonperiodic, for even R there is no contact point exist. This is the case of A053417.
Sequence A053416 addresses the case in which the center of the R-circle (R = r*n) is chosen at a lattice point instead; in that case, the number of contact points is 0 and 6 for even n > 0 and odd n > 1, respectively.
See illustrations in links.

Kival Ngaokrajang, Illustration R-circle center is at (0.5,0)
Kival Ngaokrajang, Illustration R-circle center is at (0,0)

Cf. A053416, A053417, A239074.

cp's OEIS Frontend

A127403 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at a grid point.

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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.

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A308685 The number of triangular lattice points whose Euclidean distance from the origin is less than or equal to n.

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A240501 Given circular disks of radius r in a hexagonal lattice covered by a circular disk of radius R = r*2n, if the center of the circle is chosen at the middle between two lattice points, a(n) is the number of points at which an r-circle is tangent to the R-circle.

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