# Van Eck's sequence

In recreational mathematics, **van Eck's sequence** is an integer sequence defined recursively as follows. Let *a*_{0} = 0. Then, for *n* ≥ 0, if there exists an *m* < *n* such that *a _{m}* =

*a*, take the largest such

_{n}*m*and set

*a*

_{n+1}=

*n*−

*m*; otherwise

*a*

_{n+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 *a*_{0} = 0. This can be changed such that the sequence starts with any integer.

For example:

With *a*_{0} = 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 *a*_{0} = 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 *a*_{0} = 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

This mathematics-related article is a stub. You can help EverybodyWiki by expanding it. |

This article "Van Eck's sequence" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Van Eck's sequence. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.