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Golomb sequence

Summary

In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a monotonically increasing integer sequence where an is the number of times that n occurs in the sequence, starting with a1 = 1, and with the property that for n > 1 each an is the smallest 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 the OEIS).

Examples

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.

Recurrence

Colin Mallows has given an explicit recurrence relation ${\displaystyle a(1)=1;a(n+1)=1+a(n+1-a(a(n)))}$ . An asymptotic expression for an is

${\displaystyle \varphi ^{2-\varphi }n^{\varphi -1},}$

where ${\displaystyle \varphi }$  is the golden ratio (approximately equal to 1.618034).

References

• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 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.