A361933 Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression in any order.

1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 9, 4, 2, 5, 11, 2, 2, 4, 1, 1, 5, 1, 1, 10, 2, 2, 4, 1, 1, 4, 4, 10, 10, 4, 8, 10, 10, 2, 4, 1, 2, 5, 4, 10, 10, 4, 2, 8, 8, 5, 8, 5, 13, 13, 17, 5, 13, 2, 11, 17, 10, 10, 13, 13

Offset: 1

Author

Neal Gersh Tolunsky, Mar 30 2023

Keywords

• nonn

Comments

First differs from A229037 and A309890 at a(28).

This sequence avoids all six of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).

This sequence is part of a family of variants avoiding different permutations of arithmetic progressions at indices in arithmetic progression:

- A100480 (offset 1), A006997 (offset 0): Prohibits 1,1,1 and progressions of common difference 0.

- A309890: Prohibits 1,2,3 or progressions of the form c, c+d, c+2d, for all d >= 0.

- A373111: Prohibits 1,3,2 or progressions of the form c, c+2d, c+d, for all d >= 0.

- A371457: Prohibits 2,1,3 or progressions of the form c, c-d, c+d, for all d >= 0.

- A371632: Prohibits 2,3,1 or progressions of the form c, c+d, c-d, for all d >= 0.

- A373010: Prohibits 3,1,2 or progressions of the form c, c-2d, c-d, for all d>=0.

- A373052: Prohibits 3,2,1 or progressions of the form c, c-d, c-2d, for all d>=0.

With the sequences prohibiting the six permutations above, there are a total of 64 sequences which prohibit some combination of these six permutations of an arithmetic progression. At least two more of these are in the OEIS:

- A229037 ("forest fire sequence"): Prohibits (progressions of the same general form as) 1,2,3 and 3,2,1 .

- A361933 (the present sequence): Prohibits all six permutations.

Examples

			a(28) cannot be 1 because then a(26)=5, a(27)=9, and a(28)=1 could be rearranged to form an arithmetic progression (1, 5, 9). The numbers 2-8 could also create an arithmetic progression so a(28)=9.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Index entries for non-averaging sequences
• Neal Gersh Tolunsky, Graph of the first 200000 terms

Crossrefs

Cf. A229037, A309890.

Programs

PARI

\\ See Links section.

Formula

a(n) <= (n+1)/2.