Golomb episode

The Golomb sequence (after the mathematician Solomon W. Golomb , but also known as the Silverman sequence ) is a self-generating sequence of integers, in which the number in front indicates how often it occurs in the sequence. For example, there is a 3 in the fifth position , so the 5 will be added 3 times later. ${\displaystyle n}$ ${\displaystyle a_{n}}$ ${\displaystyle n}$ ${\displaystyle (n=5)}$

construction

In the first place there is the 1, which means that it occurs exactly once. Since this condition is fulfilled at the same time, no further 1 can appear, and the second follows in the second place ( ). It follows that the 2 occurs twice in the sequence. After the existing one, another 2 is added accordingly, so that a 2 is also in the third position ( ). This means that the 3 appears twice. So the sequence up to this point is: 1,2,2,3,3. Since there is now a 3 in the fourth and fifth position, exactly 3 fours and 3 fives are added: 1,2,2,3,3,4,4,4,5,5,5. This gives you the digits 6 to 11 and you can read from them how many sixes, sevens, etc. continue the sequence. ${\displaystyle n=1}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$

This results in for the first : 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7 (sequence A001462 in OEIS ) . ${\displaystyle n}$

Formal definition

The statistician Colin Mallows found the difference equation for the next one as a mathematical description of this recursion ${\displaystyle n}$

${\displaystyle a_{n+1}=1+a_{n+1-a_{a_{n}}}({\text{für }}n>1;a_{1}=1)}$

Or in alternative notation:

${\displaystyle a(n+1)=1+a(n+1-a(a(n)))}$

example

If the first four digits of the sequence are known, the following applies to the fifth:

${\displaystyle a_{5}=a_{4+1}=1+a_{4+1-a_{a_{4}}}=1+a_{5-a_{3}}=1+a_{5-2}=1+a_{3}=1+2=3}$

${\displaystyle a_{5}=3}$, so the 5 occurs three times.

An approximation for any values ​​of can be calculated with the golden ratio (≈ 1.618): ${\displaystyle a_{n}}$ ${\displaystyle n}$ ${\displaystyle \Phi }$

${\displaystyle a_{n}\approx \Phi ^{2-\Phi }\cdot n^{\Phi -1}}$

example

${\displaystyle a_{57}\approx \Phi ^{2-\Phi }\cdot {57}^{\Phi -1}\approx 14{,}6223278\approx 15}$

${\displaystyle a_{57}\approx 15}$, ie, according to the approximation formula, the 57 appears 15 times in the sequence (actual value: 15).

See also

• Fibonacci sequence
• Conway episode

Individual evidence

1. ↑ ^{a b} OEIS: Golomb's sequence . Retrieved March 16, 2014.
2. ↑ B. Cloitre, NJA Sloane, MJ Vandermast: Numerical Analogues of Aronson's Sequence on arXiv.org . Retrieved March 16, 2014.
3. ^ OEIS: Golomb's sequence: Table of n, a (n) for n = 1..10000 . Retrieved March 16, 2014.