Van Eck's sequence
In recreational mathematics, van Eck's sequence is an integer sequence defined recursively as follows. Let a0 = 0. Then, for n ≥ 0, if there exists an m < n such that am = an, take the largest such m and set an+1 = n − m; otherwise an+1 = 0. Thus the first occurrence of an integer in the sequence is followed by a 0, and the second and subsequent occurrences are followed by the size of the gap between the two most-recent occurrences.[1]
The first few terms of the sequence are (OEIS: A181391):
- 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 ... [2]
The sequence was named by Neil Sloane after Jan Ritsema van Eck, who contributed it to the On-Line Encyclopedia of Integer Sequences in 2010.
Properties[edit]
It is known that the sequence contains infinitely many zeros and that it is unbounded.[2]
It is conjectured, but not proved, that the sequence contains every positive integer, and that every pair of non-negative integers apart from (1,1) and (n,n+1) appears as consecutive terms in the sequence.[2]
Variations[edit]
The sequence OEIS: A181391 is defined with a0 = 0. This can be changed such that the sequence starts with any integer.
For example:
With a0 = 1, OEIS: A171911:
- 1, 0, 0, 1, 3, 0, 3, 2, 0, 3, 3, 1, 8, 0, 5, 0, 2, 9, 0, 3 ...[3]
With a0 = 2, OEIS: A171912:
- 2, 0, 0, 1, 0, 2, 5, 0, 3, 0, 2, 5, 5, 1, 10, 0, 6, 0, 2, 8 ...[4]
In fact, OEIS has eight other entries, from A171911 to A171918, corresponding to the separate sequences generated with a0 = 1 to 8.
References[edit]
- ↑ "The Van Eck Sequence". June 12, 2019.
- ↑ 2.0 2.1 2.2 van Eck's sequence (A181391) at the On-Line Encyclopedia of Integer Sequences
- ↑ "A171911 - OEIS". oeis.org. Retrieved 2019-06-17.
- ↑ "A171912 - OEIS". oeis.org. Retrieved 2019-06-17.
External links[edit]
- Brady Haran and N. J. A. Sloane, Don't Know (the Van Eck Sequence) on YouTube
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