A382501 Lexicographically earliest infinite sequence of positive integers such that, for any given k, every subsequence {a(j), a(j+k), a(j+2k)} (j, k >= 1) is unique.

1, 1, 1, 2, 1, 1, 3, 1, 2, 4, 3, 1, 1, 4, 1, 3, 2, 5, 2, 4, 2, 3, 4, 1, 2, 5, 3, 2, 4, 6, 1, 3, 5, 5, 6, 1, 1, 7, 2, 3, 8, 4, 8, 7, 1, 2, 6, 5, 3, 1, 4, 3, 8, 7, 2, 8, 2, 6, 9, 1, 9, 1, 4, 6, 9, 4, 5, 9, 2, 7, 5, 7, 3, 4, 3, 10, 10, 4, 9, 1, 3, 6, 2, 5, 8, 2, 9

Offset

1,4

Comments

Every subsequence {a(n-2k), a(n-k) a(n)} with its corresponding k value (or index spacing) is unique.

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Sean A. Irvine, Java program (github)

Example

To find a(10) = 4, we first try 1. We cannot have a(10) = 1 because this would create the subsequence {1,1,1} at i = 6,8,10, which occurred before at i = 1,3,5. In both cases, k = 2, which is not allowed .
a(10) cannot be 2 because then the subsequence {1,1,2} at i = 2,6,10 would be the same as {1,1,2} at  i = 1,5,9. In both cases, k = 4.
a(10) cannot be 3 because {1,1,3} at i = 6,8,10 would be the same as the subsequence at i = 3,5,7. In both cases, k = 2.
When we try a(10) = 4, we see that none of the new subsequences formed have occurred before with the same k value. Since 4 is a first occurrence, every subsequence created is new, and although i = 6,8,10 has the same subsequence {1,1,4} as i = 2,6,10, the k value is different, which is allowed. So a(10) = 4.

Crossrefs

Cf. A364057, A382502.

Sequence in context: A029288 A238899 A187207 * A050117 A241187 A212822

Adjacent sequences: A382498 A382499 A382500 * A382502 A382503 A382504

Keyword

nonn

Author

Neal Gersh Tolunsky, Mar 29 2025

Status

approved