Golomb sequence

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 2, 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.

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 $$a(1) = 1; a(n+1) = 1 + a(n + 1 - a(a(n)))$$. An asymptotic expression for an is


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

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