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 A309797 Lexicographically earliest sequence of positive integers such that for any n > 0 there are no more than a(n) numbers k > 0 such that a(n + k) = a(n + 2*k). 4
 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 4, 2, 4, 5, 4, 1, 1, 1, 1, 3, 3, 3, 1, 6, 6, 6, 7, 5, 3, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 7, 5, 7, 5, 8, 5, 8, 8, 8, 8, 3, 1, 1, 1, 1, 9, 9, 6, 1, 1, 1, 1, 10, 3, 3, 3, 6, 6, 6, 6, 6, 7, 6, 3, 9, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The sequence is well defined as we can always extend the sequence with a number that has not yet appeared. The number 1 appears infinitely many times in the sequence: - by contradiction: suppose that m is the index of the last occurrence of 1 in the sequence, - there is no n > 0 such that n + k = m and n + 2*k = 2*m (with k > 0), - so we can choose a(2*m) = 1, QED. This sequence has connections with A003602: - here we have up to a(n) numbers k such that a(n+k) = a(n+2*k), there we have no such numbers, - for any v >= 0, let f_v be the lexicographically earliest sequence of positive integers such that there are no more than v numbers k such that f_v(n + k) = f_v(n + 2*k), - then f_v corresponds to A003602 where all but the first term have been repeated 2*v+1 times. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 Rémy Sigrist, PARI program for A309797 Rémy Sigrist, Scatterplot of the first 500000 terms FORMULA a(n) >= #{ k>0 such that a(n+k) = a(n+2*k) }. EXAMPLE The first terms, alongside the corresponding k's, are:   n   a(n)  k's   --  ----  ----------    1     1  {1}    2     1  {1}    3     1  {2}    4     1  {1}    5     2  {1, 3}    6     2  {2, 14}    7     2  {1, 2}    8     1  {1}    9     1  {1}   10     1  {4}   11     1  {1}   12     3  {2, 3, 68}   13     3  {1, 4, 22}   14     2  {1, 2} PROG (PARI) See Links section. CROSSREFS Cf. A003602, A329268 (positions of 1). Sequence in context: A029432 A073426 A232439 * A199840 A126067 A327490 Adjacent sequences:  A309794 A309795 A309796 * A309798 A309799 A309800 KEYWORD nonn,look AUTHOR Rémy Sigrist, Nov 11 2019 STATUS approved

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Last modified April 9 10:32 EDT 2020. Contains 333348 sequences. (Running on oeis4.)