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Consider j identical regular m-gons, assembled into a circular closed chain. Two neighboring polygons share an edge and two vertices, the "inner" one lying in the interior of the chain. The interior is a j-pointed star with equal edges.

n is introduced in order to partition the set of chains into finite subsets. Two neighboring star points are separated by n vertices; there the star has reflex angles. (With n=0, regular polygons are considered as stars with no reflex angles.)

For every m > 4 there exists a chain of m-gons.

A366872 gives the number of row elements.

This sequence is interconnected with A383169. For each n there are finitely many pairs (m,j) for j m-gons building closed chains. m are given by T(n,k) and the corresponding j are given by A383169(n,k).

j = 2 + (8+4n)/(m-4-2n).

m = 4 + 2n + (8+4n)/(j-2).

These two equations allow a computation of T(n,k) and A383169(n,k) from each other, see Formula.

Consider j identical regular m-gons, assembled into a circular closed chain. Two neighboring polygons share an edge and two vertices, the "inner" one lying in the interior of the chain. The interior is a j-pointed star with equal edges.

n is introduced in order to partition the set of chains into finite subsets. Two neighboring star points are separated by n vertices; there the star has reflex angles. (With n=0, regular polygons are considered as stars with no reflex angles.)

For every j > 2 there exists a chain with exactly j polygons.

A366872 gives the number of row elements.

The descending order in the definition was chosen with respect to the interconnection with A383168. For each n there are finitely many pairs (m,j) for j m-gons building closed chains. j are given by T(n,k) and the corresponding m are given by A383168(n,k).

m = 4 + 2n + (8+4n)/(j-2).

j = 2 + (8+4n)/(m-4-2n).

These two equations allow a computation of T(n,k) and A383168(n,k) from each other, see Formula.

Partial row sums of A163353.

For covers (collections without an empty set) see A369950.

For disjoint collections see A381682.

For disjoint covers see A102661.

Partial row sums of A256894.

For disjoint covers (collections without an empty set) see A102661.

For non-disjoint collections see A381683.

For non-disjoint covers see A369950.

Number of n-tuples containing all elements of [m] with a unique last element.

Consider an urn with m balls of pairwise different colors. T(n,m) is motivated by the probability p(n,m) for exactly n draws with replacement needed to obtain all colors; p(n,m)=T(n,m)/m^n. - With m fixed and n running, p(n,m) is a probability distribution. The expected number of draws needed to obtain all colors is Sum_{j=1..m} m/j. (Expected value provided by M. Shackleford.)

Numbers appearing exactly once in a Pythagorean triple and as the smallest number in this triple.

Subsequence of A292762.

Inserting 4 as second term gives A374846.

Positions of the ones in A046081.

With the exception a(2) = 4, the terms are given by A374845, thus providing a simple formula for the sequence.

Partial row sums of A055154.

Also, number of k-covers of [n] allowing for empty subsets.

Amendments by Manfred Boergens, Mar 09 2025: (Start)

For covers which may include one empty set see A381683.

For disjoint covers see A102661.

For disjoint covers which may include one empty set see A381682. (End)

Consider j identical regular polygons, assembled into a circular closed chain. Two neighboring polygons share an edge and two vertices, the "inner" one lying in the interior of the chain. The interior is a j-pointed star with equal edges.

n is introduced in order to partition the set of chains into finite subsets. Two neighboring star points are separated by n vertices; there the star has reflex angles. (With n=0, regular polygons are considered as stars with no reflex angles.)

Geometrical reasoning shows that for each n there are finitely many (not zero) chains with the described properties.

a(n) is the number of these chains and equals d(8+4n), the number of divisors of 8+4n.

For every m > 4 there exists a chain of m-gons. The possible m for each n are given by A383168.

For every j > 2 there exists a chain with exactly j polygons. The possible j for each n are given by A383169.

The sequence deals with sets of n points with pairwise different distances. The randomness in the definition 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 simple digraph; the vertices of G are n arbitrarily 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). Let b(n) be the minimal number of points receiving arrowheads in any such graph. a(n) is the best upper bound yet known for b(n).

A261953(n) for n >= 2 can be seen as an "inverse" to a(n).

a(n) is built by constructing G with n points and m nearest neighbors, m chosen as minimal as possible, then defining a(n)=m.

The start is a(n)=2 for n <= 9 and a(n)=3 for n=10,11,12. We call the pairs (n,m)=(9,2) and (n,m)=(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).

If (n0,m-1) and (n1,m) are anchor pairs then a(n')=m for n0 < n' <= n1.

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