A053417 -id:A053417 - cp's OEIS frontend

A053459 Open disk numbers (version 4): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (1/2,0).

0, 0, 4, 8, 14, 22, 30, 42, 60, 74, 92, 108, 130, 148, 178, 206, 230, 262, 288, 324, 364, 400, 442, 480, 522, 562, 614, 662, 712, 764, 812, 868, 922, 988, 1050, 1106, 1176, 1234, 1312, 1376, 1452, 1528, 1598, 1678, 1750, 1838, 1920, 2006, 2092, 2172, 2266

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 13 2000

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

Cf. A053417, A053456, A053457, A053458.

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

A053459 Open disk numbers (version 4): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (1/2,0).

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