A239073 -id:A239073 - cp's OEIS frontend

A053417 Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).

0, 2, 4, 10, 14, 24, 30, 48, 60, 76, 92, 110, 130, 154, 178, 208, 230, 264, 288, 330, 364, 406, 442, 482, 522, 564, 614, 664, 712, 766, 812, 874, 922, 990, 1050, 1112, 1176, 1240, 1312, 1382, 1452, 1530, 1598, 1684, 1750, 1840, 1920, 2008, 2092, 2182, 2266

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000

Equivalently, number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen at the middle between two lattice points. - Hugo Pfoertner, Jan 07 2007

Same as above but "number of disks (r = 1)" instead of "number of points". a(2^n - 1) = A239073(n), n >= 1. See illustration in links. - Kival Ngaokrajang, Apr 06 2014

H. v. Eitzen, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A053411, A053414, A053415, A053416, A053417.
Cf. A053479, A125851, A125852.
Cf. A239073.

a[n_] := Sum[dj = Sqrt[Abs[4*n^2 + 6*i - 3*i^2 - 3]]/4; j1 = (1 - 2*i)/4 - dj // Floor; j2 = (1 - 2*i)/4 + dj // Ceiling; Sum[ Boole[i^2 - i - j/2 + i*j + j^2 + 1/4 <= n^2/4], {j, j1, j2}], {i, -n - 1, n + 3}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

a(n)/(n/2)^2 -> Pi*2/sqrt(3).

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A239074 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 number of points at which a small circle is tangent to the big circle.

Original entry on oeis.org

2, 2, 6, 2, 6, 6, 6, 2, 18, 6, 2, 18, 6, 18, 54, 2, 6, 54, 6, 6, 90, 2, 2, 54, 54, 18, 54, 18, 6, 162, 6, 2, 18, 6, 18, 1458, 18, 18, 162, 18, 2, 810, 6, 18, 1458, 2, 6, 486, 18, 162, 486, 54, 6, 486, 18, 54, 162, 18, 2, 4374, 6, 18, 2430, 6, 54, 162, 18, 18, 18, 54

Offset: 1

Kival Ngaokrajang, Apr 06 2014

nonn

The construction rule is: (1) Start with a unit circle (big R-circle). (2) Pack circles at radius 1/2^n (small r-circles) on the diameter line of the big circle. (3) Pack small circles in the rows above and below the row packed in the previous step, maintaining a hexagonal packing pattern. The number of small circles in any row is limited so that the circumference of the last small circle does not cross (but is allowed to contact) the circumference of the big circle. (4) Repeat process to the top and bottom rows.
The contact points are the points where the circumference of a small circle contacts the circumference of the big circle, i.e., they are mutually tangent.
See illustration in links.
Also, the number of integer solutions to the equation (2^n-1)^2 = 3*x^2 + y^2. - Andrew Howroyd, May 27 2018

			n=9 (see the link): In the first quadrant, shown there, there are 4 touching points with the large circle for rows x > 0, namely for the rows 52, 132, 280 and 292. With the trivial 2 touching points with the large circle for the row x=0 this adds to the total number 2 + 4*4 = 18 = a(9). - _Wolfdieter Lang_, Apr 06 2014

Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Illustration for n = 9

Cf. A033716, A239073, A239206.

a[1] = 2; a[n_] := Module[{f = FactorInteger[2^n - 1]}, 2*Product[If[Mod[ f[[i, 1]], 3] == 1, 2*f[[i, 2]] + 1, 1] , {i, 1, Length[f]}]];
Array[a, 70] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

a(n) = {my(f=factor(2^n-1)); 2*prod(i=1, #f~, if(f[i, 1]%3==1, 2*f[i, 2]+1, 1))} \\ Andrew Howroyd, May 27 2018

a(n) = 2 + 4*A(n), n >= 1, with A(n) the number of integer solutions for x(n,j) = sqrt((2^n-1)^2 + 3*j^2), for j = 1, 2, ..., floor((2^n-1)/sqrt(3)). R = 2^n and r=1 (small radius) was used here. - Wolfdieter Lang, Apr 07 2014

a(n) = A033716((2^n - 1)^2). - Andrew Howroyd, May 27 2018

Corrected and extended by Wolfdieter Lang, Apr 06 2014

a(26)-a(70) from Andrew Howroyd, May 27 2018

A239206 a(n) is the total number of rows of circles of radius r packing into a circle of radius R, where r = R/2^n.

Original entry on oeis.org

1, 1, 3, 9, 17, 35, 73, 147, 295, 591, 1181, 2363, 4729, 9459, 18917, 37837, 75673, 151347, 302697, 605395, 1210791, 2421581, 4843163, 9686329, 19372659, 38745319, 77490641, 154981281, 309962565, 619925129, 1239850261, 2479700523, 4959401047, 9918802097

Offset: 0

Kival Ngaokrajang, Mar 12 2014

nonn

See illustration in link for construction rule.
From Wolfdieter Lang, Apr 01 2014: (Start)
Call in the above mentioned illustration (link) the horizontal layers with the y-axis (center as origin) intersecting a circle B-layers and the other ones A-layers. Consider first only the upper half, with the A-layer y=0 treated separately. The largest k value for B-layers  with yB(k) = sqrt(3)*(2*k + 1), k >= 0, is kBmax(n) = floor((2^n-1-sqrt(3))/(2*sqrt(3))), n>=2. Similarly, the largest k value for A-layers with yA(k) = sqrt(3)*2*k, k >= 0, is kAmax(n) = floor(sqrt((2^n-1)^2-1)/(2*sqrt(3))), n >= 1. For n=0 one has the large disk. If the top level is of B-type then 2*kBmax(n) + 1 >  2*kAmax(n). In this case the total number of layers (A or B) is 4*kBmax(n) + 3, n >= 2. In the other case (top level is of the A-type) it is 1 + 4*kAmax(n), n >= 1. With a(0) = 1 Kival Ngaokrajang's formula given below, which is a(n) = 2*floor((2^n-1)/sqrt(3)) + 1, seems to fit these numbers. (End)

Kival Ngaokrajang, Illustration of initial terms

Cf. A239073, A239074.

rcr[n_]:=Module[{r=1/2^n,c=Cos[Pi/6]},2*Floor[(1-r)/(2*r*c)]+1]; Array[ rcr,40,0] (* Harvey P. Dale, Dec 23 2014 *)

a(n) = 1 + 2*sqrtint((2^n-1)^2\3); \\ Andrew Howroyd, May 27 2018
(Small Basic)
For n = 0 To 50
  r = 1/Math.Power(2,n)
  a = 2*math.Floor((1-r)/(2*r*math.Cos(Math.Pi/6))) + 1
  TextWindow.Write(a+", ")
EndFor

a(n) = 2*floor((1-r)/(2*r*cos(Pi/6))) + 1, where r = 1/2^n.

a(n) = 2*floor(sqrt((2^n-1)^2/3)) + 1. - Andrew Howroyd, May 27 2018

cp's OEIS Frontend

A053417 Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).

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A239074 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 number of points at which a small circle is tangent to the big circle.

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A239206 a(n) is the total number of rows of circles of radius r packing into a circle of radius R, where r = R/2^n.

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Author

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Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula