cp's OEIS Frontend

See a comment on V-V and V-S at A249246.

There are a total of 16 combinations as shown in the table below:

+-------------------------------------------------------+

| Even n-th version V-V S-V V-S S-S |

+-------------------------------------------------------+

| Odd n-th version |

| V-V A008486 A248969 A261951 A261952 |

| S-V A261950 A008486 A008486 A261956 |

| V-S A249246 A008486 A008486 A261957 |

| S-S a(n) A261954 A261955 A008486 |

+-------------------------------------------------------+

Note: V-V = vertex to vertex, S-V = side to vertex,

V-S = vertex to side, S-S = side to side.

From Manfred Boergens, Sep 21 2021: (Start)

For finite sets of random points in the real plane with exactly n nearest neighbors, a(n) for n >= 2 is a lower bound for the maximal number of points. Conjecturally, a(n) equals this number.

The randomness provides for pairwise different distances with probability = 1.

A point A is called a nearest neighbor if there is a point B with smaller distance to A than to any other point C.

In graph theory terms: Let G be a finite simple digraph; the vertices of G are arbitrary placed points in R^2 with pairwise different distances; the edges of G are arrows joining each point (tail end) to its nearest neighbor (head end). If G contains exactly n nearest neighbors and b(n) is the maximal number of points in any such graph then a(n) is the best lower bound known as yet for b(n).

a(n) for n >= 2 can be seen as an "inverse" to A347941.

a(n) is built by constructing G with m points and n nearest neighbors, m chosen as maximal as possible, then defining a(n)=m. The start is a(2)=9 and a(3)=12. We call the pairs (m,n)=(9,2) and (m,n)=(12,3) "anchor pairs" and proceed to bigger n by combining graphs with these anchor pairs to bigger graphs. So the next anchor pairs are (18,4), (21,5) and (27,6).

We conjecture that a(n) is optimal. This claim is true if the following assumptions hold:

- The anchor pairs (9,2) and (12,3) are optimal.

- All bigger anchor pairs (m,n) are constructed by combining copies of (9,2) if n is even and adding one (12,3) if n is odd.

(End)