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A361933
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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.
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8
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) <= (n+1)/2.
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EXAMPLE
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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.
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PROG
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(PARI) \\ See Links section.
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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