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In mathematics, the **Golomb sequence**, named after Solomon W. Golomb (but also called **Silverman's sequence**), is a monotonically increasing integer sequence where *a _{n}* is the number of times that

*a*_{1} = 1

Therefore, 1 occurs exactly one time in this sequence.

*a*_{2} > 1

*a*_{2} = 2

2 occurs exactly 2 times in this sequence.

*a*_{3} = 2

3 occurs exactly 2 times in this sequence.

*a*_{4} = *a*_{5} = 3

4 occurs exactly 3 times in this sequence.

5 occurs exactly 3 times in this sequence.

*a*_{6} = *a*_{7} = *a*_{8} = 4

*a*_{9} = *a*_{10} = *a*_{11} = 5

etc.

Colin Mallows has given an explicit recurrence relation . An asymptotic expression for *a _{n}* is

where is the golden ratio (approximately equal to 1.618034).

- 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.

- Python code for Golomb Sequence