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 A359866 a(n) is the number of k > 0 such that n-1-2*k >= 0 and a(n-1-2*k) >= a(n-1-k) >= a(n-1). 2
 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 4, 0, 3, 0, 3, 1, 0, 7, 0, 6, 0, 7, 0, 6, 0, 7, 1, 0, 9, 0, 10, 0, 8, 0, 10, 0, 9, 0, 10, 1, 0, 11, 0, 14, 0, 13, 0, 14, 0, 12, 0, 13, 1, 1, 3, 0, 17, 0, 17, 0, 18, 0, 17, 0, 18, 0, 17, 1, 2, 3, 0, 21, 0, 23, 0, 22, 0, 23, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS This sequence is unbounded: - by contradiction: suppose that a(n) < M for some fixed M, - then, by Van der Waerden's theorem, we have an arithmetic progression of 2*M+1 indices where the sequence has the same value: say a(m) = a(m + k*r) for k = 0..2*M with m >= 0 and r > 0, - this would imply that a(m + 2*M*r + 1) >= M, a contradiction. This sequence has infinitely many 0's (if a(m) < a(n) for any m < n, then a(n+1) = 0). LINKS Rémy Sigrist, Table of n, a(n) for n = 0..10000 Rémy Sigrist, Scatterplot of the first 250000 terms Rémy Sigrist, C program Wikipedia, Van der Waerden's theorem. EXAMPLE The first terms, alongside the corresponding k's, are: n a(n) k's -- ---- ------------ 0 0 {} 1 0 {} 2 0 {} 3 1 {1} 4 0 {} 5 1 {2} 6 0 {} 7 1 {2} 8 1 {2} 9 0 {} 10 3 {1, 2, 3} 11 0 {} 12 4 {2, 3, 4, 5} PROG (C) See Links section. CROSSREFS Cf. A359867. Sequence in context: A272474 A340426 A308717 * A297217 A218859 A363028 Adjacent sequences: A359863 A359864 A359865 * A359867 A359868 A359869 KEYWORD nonn,look AUTHOR Rémy Sigrist, Jan 16 2023 STATUS approved

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Last modified July 18 06:54 EDT 2024. Contains 374377 sequences. (Running on oeis4.)