Conway episode

The Conway sequence is a mathematical sequence named after the British mathematician John Horton Conway . It was first published in 1986 by John Conway. ( Lit .: Conway, 1986).

The Conway episode is very often found as a puzzle . Usually the first few parts of the episode are revealed and the candidate is asked to continue the episode. Due to the rather unusual definition of the episode, this has some headache.

Because of the way she was educated , Conway called her the Audioactive Series.

definition

The terms of the sequence are defined recursively in a way that is quite curious for mathematics . The members are not to be viewed in the actual sense as numbers in the decimal system , but merely as sequences of digits, from the description of which the successor sequence of digits is determined. The starting value is always a positive natural number (or any sequence of digits), usually . To determine the next element, you determine the length of the blocks of identical digits in the previous number and write the frequency and digit for each block one after the other. The number so written is the next term in the sequence. ${\ displaystyle d}$ ${\ displaystyle d = 1}$

Illustration of the definition for d = 1

n	predecessor	n-th successor
1	-	1
2	a one	11
3	two ones	21st
4th	a two, a one	1211
5	a one, a two, two ones	111221
6th	three ones, two twos, one one	312211
7th	a three, a one, two twos, two ones	13112221
...	etc.
70	...	Decimal notation has 179,691,598 digits.

Conway sequence for different initial values

d	line	OEIS link
1	1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...	Follow A005150 in OEIS
2	2, 12, 1112, 3112, 132112, 1113122112, 311311222112, ...	Follow A006751 in OEIS
3	3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ...	Follow A006715 in OEIS
...
22nd	22, 22, 22, ...	( Episode is stationary )

Mathematical properties

The length of the sequence diverges for all starting values with the exception of 22 and grows very quickly. The decimal representation of the 70th sequence member for already has 179,691,598 digits. Asymptotically , the length of the sequential terms increases with speed . This is the so-called Conway constant . ${\ displaystyle d}$ ${\ displaystyle \ infty}$ ${\ displaystyle d = 1}$ ${\ displaystyle \ Theta (\ lambda ^ {n})}$ ${\ displaystyle \ lambda \ approx 1 {,} 303577 \ ldots}$

If the starting value only contains the digits 1, 2 and 3 and all sequences of the same digits are no more than three digits long, all other elements of the Conway sequence also consist only of the digits 1, 2 and 3, whereby the digit sequence ... 333 ... occurs.

Web links

Conway episode in MathWorld: Look and Say Sequence (English)
Conway sequence in the On-Line Encyclopedia of Integer Sequences (English)
John Conway on this sequence (English)

literature

John Horton Conway : The Weird and Wonderful Chemistry of Audioactive Decay. Eureka 46, 1986, pp. 5-18
Clifford Stoll : Cuckoo's Egg - The high-ranking NSA employee Robert Morris gives Clifford Stoll the first parts of the Conway sequence as a number puzzle that Stoll cannot solve.