Maurice Clerc - cp's OEIS frontend

User: Maurice Clerc

Maurice Clerc has authored 2 sequences.

A376215 Maximum number of non-overlapping discs with a diameter of 1 that can fit within an n×n square, using hexagonal packing or square packing, whichever is more efficient.

1, 4, 9, 16, 25, 36, 49, 68, 85, 105, 126, 150, 175, 216, 247, 279, 314, 350, 389, 429, 492, 538, 585, 635, 686, 740, 822, 880, 941, 1003, 1068, 1134, 1203, 1307, 1380, 1456, 1533, 1613, 1694, 1817, 1904, 1992, 2083, 2175, 2270, 2366, 2511, 2613, 2716, 2822, 2929, 3039

Offset: 1

Maurice Clerc, Sep 15 2024

Maximum error on the first 99 terms = 3.3%. (See the Specht's site).
The first row of discs is placed on the base of the square. Then hexagonal packing is applied, row by row, as much as possible. We find a number N of discs. If n^2>N then the whole packing is replaced by the square one. Example: n=3 => N=8 but since n^2=9 we can indeed place 9 discs.
Note: useful only for n=2, 3, 4, 5, 6, 7.

E. Specht, The best known packings of equal circles in a square

Cf. A084617.

from math import isqrt
def A376215(n): return max(n**2,n*(m:=1+isqrt(((n-1)**2<<2)//3))-(m>>1)) # Chai Wah Wu, Nov 06 2024

a(n) = max(n^2,n*floor(1+2*(n-1)/sqrt(3)) - floor(floor(1+2*(n-1)/sqrt(3))/2));

A376066 Minimum number of unit squares needed to cover the circumference of a circle of radius n.

Original entry on oeis.org

4, 9, 14, 18, 23, 27, 32, 36, 40, 45, 49, 54, 58, 63, 67, 72, 76, 80, 85, 89, 94, 98, 103, 107, 112, 116, 120, 125, 129, 134, 138, 143, 147, 152, 156, 160, 165, 169, 174, 178, 183, 187, 192, 196, 200, 205, 209, 214, 218, 223, 227, 232, 236, 240, 245, 249, 254, 258, 263, 267, 272, 276, 280, 285, 289, 294, 298, 303, 307, 311

Offset: 1

Maurice Clerc, Sep 08 2024

For n>=2, a unit square covers the most circumference when it has two diagonally opposite corners on the circumference, forming a chord of length sqrt(2).

A simple upper bound a(n) <= u(n) = ceiling(2*Pi*n/sqrt(2)) would be by sqrt(2) arcs instead of chords, and which is bigger at for instance a(70) = 311 < u(70) = 312 (see A376207).

Cf. A376207.

a(n) = ceiling(Pi/arcsin(sqrt(2)/(2*n))).

cp's OEIS Frontend

User: Maurice Clerc

A376215 Maximum number of non-overlapping discs with a diameter of 1 that can fit within an n×n square, using hexagonal packing or square packing, whichever is more efficient.

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