# Godel Escher Bach: Interesting number sequence

• DaveC426913
In summary, the conversation discusses an interesting algorithm called the Collatz conjecture, which involves repeatedly halving or tripling a number until it reaches 1. The conversation also mentions some patterns and connections to recursion theory and the halting problem. Erdos is quoted as saying that mathematics is not yet ready for such problems. The conversation also brings up the idea of a "hailstone function" and its relation to the Collatz conjecture.
DaveC426913
Gold Member
I've been re-reading https://www.amazon.com/dp/0465026567/?tag=pfamazon01-20 and came across an interesting algorithm.

If it is even, halve it.
If it is odd, triple it and add 1.
Repeat until the number reaches 1.
Count # of steps it took.

Code:
Number/Count
1: (1)                  count: 0
2: (2 1)                count: 1
3:  (3 10 5 16 8 4 2 1) count: 7
etc.
This acts very well-behaved for the most part; the count never goes higher than 23

Until you get to 27...
Code:
Number/Count
25: (25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 ) count: 23
26: (26 13 40 20 10 5 16 8 4 2 1 ) count: 10
27:  (27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 ) count: 111
then it jumps to a whopping 111.

I've been doing some perfunctory analysis on this (including a little javascript snippet to generate the #s for me (don't get me wrong, the JavaScript came only AFTER I did it all by hand up to 27 (including triple-checking my #s (I was stranded on a plane and had nothing to keep me busy except a pencil and a barf bag with lots of white space on it...(BTW, did I mention that the book is all about recursion?))))) and have found some tantalizing patterns.

Has anyone seen this algorithm before?

Last edited by a moderator:
Oooh! Thank you!

Little did I know, when I was scribbling on the back of that barf bag, that I was generating the first few pixels of a Mandelbrot-esque set!

If I state the "hailstone function" as,

f(1) = 1, f(2n) = f(n), f(2n+1)=(3(2n+1)+1)/2

In the language of recursion theory, this is a partial function. The conjecture is whether this is total, correct? Is this related to the halting problem in some way? Is there a theorem stating whether all recursively defined functions of a certain type are total/partial?

## 1. What is the significance of the number sequence found in Godel Escher Bach?

The number sequence found in Godel Escher Bach, also known as the Hofstadter sequence, is significant because it is a self-referential sequence that demonstrates the concept of recursion and the emergence of complex patterns from simple rules. It is also related to the work of mathematician Kurt Godel, artist M.C. Escher, and composer Johann Sebastian Bach, as explored in the book by Douglas Hofstadter.

## 2. How is the number sequence generated?

The Hofstadter sequence is generated by recursively applying two simple rules: each number is equal to the number that is two positions before it, minus the number that is one position before it.

## 3. Can the number sequence be extended infinitely?

Yes, the Hofstadter sequence can be extended infinitely by continually applying the two recursive rules mentioned above. However, as the numbers get larger, it becomes increasingly difficult to calculate and predict the values.

## 4. Are there any practical applications of the Hofstadter sequence?

Although the Hofstadter sequence may not have any direct practical applications, it has been used in mathematical and computer science research to explore the concept of recursion and self-referential systems. It has also inspired other fields such as linguistics and music theory.

## 5. Are there any patterns or properties of the number sequence?

Yes, there are several interesting patterns and properties of the Hofstadter sequence, including the fact that it contains prime numbers, its relationship to the Fibonacci sequence, and the existence of a "golden ratio" between consecutive numbers. However, many aspects of the sequence are still not fully understood and continue to be studied by mathematicians and scientists.

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