### - Art Gallery -

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.

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 ... [1]

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

It is known that the sequence contains infinitely many zeros and that it is unbounded.[1]

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.[1]
Variations

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 ...[2]

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 ...[3]

In fact, OEIS has eight other entries, from A171911 to A171918, corresponding to the separate sequences generated with a0 = 1 to 8.
References

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.

Brady Haran and N. J. A. Sloane, Don't Know (the Van Eck Sequence) on YouTube