Let’s call the sequence *V* so that *V*(*n*) is the n-th term of the sequence.

Let’s define *V*(1) = 0

In order to figure out the next term *V*(*n* + 1) you check whether there exists *m* < *n* such that *V*(*m*) = *V*(*n*) (in other words, whether *V*(*n*) has appeared before in the sequence)

If such *m* exists, then *V*(*n* + 1) = *n* − *m* (in other words, count down how many steps ago such number appears).

If no such *m* exists, then *V*(*n* + 1) = 0.

The resulting sequence is called **Van Eck’s sequence**, named after Jan Ritsema van Eck, who first submitted it to The On-Line Encyclopedia of Integer Sequences at A181391.

Ony a few facts are known about this sequence (Sloane and others 2013):

- It contains infinitely many zeroes,
- After
*n* terms, there are sometimes terms around *n*
- As per the definition, it is obvious that
*V*(*n*) cannot be larger than *n*

Other than that not much is known. It is conjectured that every positive integer appears at least once in the sequence, but a definitive proof does not exist at the time.

**Andy’s Brevity Quest 2019** (229 w) → Across The Universe

# References

Sloane, Neil JA, and others. 2013. “The on-Line Encyclopedia of Integer Sequences.” *Ann. Math. Inform* 41: 219–34.