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A337747 Maximal number of 4-point circles passing through n points on a plane. 1
0, 0, 0, 1, 1, 3, 6, 12, 14, 22 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

This is a variant of the orchard-planting problem that uses circles instead of straight lines.

The maximal number of 3-point circles passing through n points on a plane is binomial(n,3). Given an arrangement of n points in general position, any choice of three points defines a circle. - Peter Kagey, Oct 05 2020

Paul Panzer provides upper and lower bounds:

a(n) <= floor[n*(n-1)*(n-2)/24].

a(n) >= 2 + n*[(n-2)*(n-2)+4]/32 for n == 0 mod 4 and n >= 8.

a(n) >= 2 + (n-1)*[(n-1)*(n-5)+16]/32 for n == 1 mod 4 and n >= 9.

a(n) >= 2 + n*(n-2)*(n-2)/32 for n == 2 mod 4 and n >= 10.

a(n) >= 2 + (n-1)*[(n-3)*(n-3)+16]/32 for n == 3 mod 4 and n >= 11.

LINKS

Table of n, a(n) for n=1..10.

Dmitry Kamenetsky, Best known solutions for n <= 13.

Dmitry Kamenetsky, Orchard planting problem, Puzzling StackExchange, August 2020.

Dmitry Kamenetsky, General orchard planting problem, Puzzling StackExchange, September 2020.

EXAMPLE

See examples in links.

CROSSREFS

Cf. A003035 (the original orchard problem), A006065.

Sequence in context: A232485 A232486 A077152 * A067759 A331068 A310122

Adjacent sequences:  A337744 A337745 A337746 * A337748 A337749 A337750

KEYWORD

nonn,more,nice

AUTHOR

Dmitry Kamenetsky, Sep 17 2020

STATUS

approved

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Last modified February 26 11:15 EST 2021. Contains 341631 sequences. (Running on oeis4.)