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Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.

A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.

It is conjectured that every positive number appears.

It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.

It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.

[The preceding comments are from Geoffrey Caveney's emails.]

From Michael De Vlieger, Aug 15 2025: (Start)

There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.

There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.

Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

A k-chain is a planar graph consisting of a continuous path made up of k straight segments. T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-chains.

The array is almost symmetric: the difference between T(k,n) and T(n,k) is 2*|k-n| (which is exactly the difference between the numbers of infinite regions). All the rows and columns satisfy the recurrence u(n) = 3*u(n-1) - 3*u(n-2) + u(n-3).

The sixth and later columns consist of all 9's, and so the antidiagonals beyond that point also consist of all 9's.

Heraclitus (circa 500 BCE) observed that no man can step in the same river twice.

The Heraclitus transform H(S) of a sequence S is formed by starting at 0, and moving s steps to the left or right, where s is any element of S, never visiting any number twice, and moving as close to 0 as possible. In case of a tie, move to the positive term.

The present sequence is the Heraclitus transform of the triangular numbers A000217. For the squares, see A377091. Conjecture: both H(A000217) and H(A000290) contain every (positive or negative) integer. In fact it appears that this property holds whenever S is a monotonically strictly increasing sequence starting with 1. It does not hold for H(A000012), which is A001477.

A "peerless" tree is an unlabeled, unrooted tree (as in A000055) with the property that if two nodes are joined by an edge then these nodes have different degrees.

Victor S. Miller reports that this sequence was first proposed on Project Euler.

Comment from Brendan McKay, May 01 2025 (Start)

The enumeration could be extended by the following argument.

If the tree has a unique centroid (not center!) then removing the centroid gives rooted subtrees of size less than n/2. If there are two centroids, they are adjacent and removing that edge gives two rooted subtrees with exactly n/2 vertices.

Start by making all rooted trees up to n/2 vertices which have no adjacent vertices of the same degree, not counting adjacencies of the root. Then classify them according to which degrees the root can be increased to without violating this condition for edges adjacent to the root.

With this information the counts for n vertices can be reconstructed. In this way getting up past 60 vertices should be possible. (End)

This sequence forms the left-most column of A383448.

It is conjectured (see A377090) that every positive integer appears exactly once either here or in A383445.

It is conjectured (see A377090) that every positive integer appears exactly once either here or in A383446.

It is a strong conjecture that every integer appears in A377091, so it is unlikely there will be a second 0 term.

Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 2.

See A382154 for further information.