Golomb sequence

In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a non-decreasing integer sequence where a_n is the number of times that n occurs in the sequence, starting with a₁ = 1, and with the property that for n > 1 each a_n is the unique integer which makes it possible to satisfy the condition. For example, a₁ = 1 says that 1 only occurs once in the sequence, so a₂ cannot be 1 too, but it can be, and therefore must be, 2. The first few values are

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 (sequence A001462 in OEIS).

a₁ = 1
Therefore 1 occurs exactly one time in this sequence.

a₂ > 1
a₂ = 2

2 occurs exactly 2 times in this sequence.
a₃ = 2

3 occurs exactly 2 times in this sequence.

a₄ = a₅ = 3

4 occurs exactly 3 times in this sequence.
5 occurs exactly 3 times in this sequence.

a₆ = a₇ = a₈ = 4
a₉ = a₁₀ = a₁₁ = 5

etc.

Colin Mallows has given an explicit recurrence relation a(1) = 1; a(n+1) = 1 + a(n + 1 - a(a(n))). An asymptotic expression for a_n is

\( \varphi^{2-\varphi}n^{\varphi-1}, \)

where φ is the golden ratio.

References

Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. pp. 10,256. ISBN 0-8218-3387-1. Zbl 1033.11006.
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. Section E25. ISBN 0-387-20860-7. Zbl 1058.11001.

External links

Python code for Golomb Sequence

Mathematics Encyclopedia