In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a monotonically increasing integer sequence where a<sub>n</sub> is the number of times that n occurs in the sequence, starting with a<sub>1</sub> = 1, and with the property that for n > 1 each a<sub>n</sub> is the smallest positive integer which makes it possible to satisfy the condition. For example, a<sub>1</sub> = 1 says that 1 only occurs once in the sequence, so a<sub>2</sub> cannot be 1 too, but it can be 2, and therefore must be 2. The first few values are
a<sub>1</sub> = 1 <br> Therefore, 1 occurs exactly one time in this sequence.
a<sub>2</sub> > 1 <br> a<sub>2</sub> = 2
2 occurs exactly 2 times in this sequence. <br> a<sub>3</sub> = 2
3 occurs exactly 2 times in this sequence.
a<sub>4</sub> = a<sub>5</sub> = 3
4 occurs exactly 3 times in this sequence. <br> 5 occurs exactly 3 times in this sequence.
a<sub>6</sub> = a<sub>7</sub> = a<sub>8</sub> = 4 <br> a<sub>9</sub> = a<sub>10</sub> = a<sub>11</sub> = 5
etc.
Colin Mallows has given an explicit recurrence relation . An asymptotic expression for a<sub>n</sub> is
where is the golden ratio (approximately equal to 1.618034).