# BrainFuck part 5 - Math sequences

DPAmar

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## Collatz sequence

Collatz sequence is defined by

- C(0) is a parameter
- C(n+1) =
- C(n)/2 if C(n) is even
- 3 * C(n) + 1 if odd

Though it has not been proved yet, the sequence is supposed to end up with ..., 4, 2, 1 and then loop 4, 2, 1, ...

Let's read an integer N and write the full Collatz sequence (comma separated) until 1 is reached

## Let's start

- Memory: empty
- Cursor: first cell
- Input: a decimal number N

## Process

- First, generate comma (code 44), to print it when needed
- Then, read integer N
- Decrease N by 1
- while N is not null (i.e. original N is not 1)
- Increase N (regenerate it)
- Copy / print N and print comma
- Try to divide N by 2
- Decrease by N
- Set else flag
- If N not null : decrease by 2, clear else flag, update quotient
- Loop until N is not null

- Then, according to else flag
- unset : N was even, and quotient N/2 is our next value
- set : N was odd, N = 2k+1, quotient is k and next value is 3N+1 = 6k+4
- In this case, multiply k by 3 : 3k
- Then add 2: 3k + 2
- And multiply by 2 : 6k+4

- in all cases, we have our next value
- copy as new N and decrease by 1 (we loop on N-1)

- Loop
- Print 1 (the final element)

## Code

```
>++++[-<+++++++++++>] ( 44 = 4 x 11)
>,[>++++++[-<-------->]>+++++++++[- read an integer
<<<[->+>+<<]>>[-<<+>>]>]<<[-<+>],] ** part 2 **
<-[ while N is not 1
+ regenerate N
[->+>+<<]>[-<+>]>[>>>>++++++++++<< copy / print N
<<[->+>>+>-[<-]<[->>+<<<<[->>>+<<< ** part 2 **
]>]<<]>+[-<+>]>>>[-]>[-<<<<+>>>>]< ** part 3 **
<<<]<[>++++++[<++++++++>-]<-.[-]<] ** part 4 **
<<.> print comma
[->+<[->->>+<<] try to divide by 2 with an else flag
>[- N is null and flag is set: N is odd
>>[-<+++>]< compute 3k
++[->++<] then 3k plus 2 and finally 6k plus 4 which is 3N plus 1
]<<] division loop
>>>[-<<<+>>>]<<<- move new N and decrease by one
] loop
<+++++. print '1' (code 49, 44 plus 5)
```

## Minified version

```
>++++[-<+++++++++++>]>,[>++++++[-<-------->]>+++++++++[-<<<[
->+>+<<]>>[-<<+>>]>]<<[-<+>],]<-[+[->+>+<<]>[-<+>]>[>>>>++++
++++++<<<<[->+>>+>-[<-]<[->>+<<<<[->>>+<<<]>]<<]>+[-<+>]>>>[
-]>[-<<<<+>>>>]<<<<]<[>++++++[<++++++++>-]<-.[-]<]<<.>[->+<[
->->>+<<]>[->>[-<+++>]<++[->++<]]<<]>>>[-<<<+>>>]<<<-]<+++++.
```

## Final state

- Memory: 49, 0, 0, 0
- Cursor: first cell
- Input: read (empty)
- Output: Collatz sequence, start from N, comma separated

*Note*: again, this is a valid sequence, modulo 256. And it does not only means that input needs to be less than 256, but that any item within sequence must be too.

Ex:

- any odd number from 87 leads to a value greater than 256
- any even number that doubles an odd value from 87 (i.e. all the 4n+2 with n >= 44) fails as well
- ...
- even worse : 85 is odd, next value is 256 = 0 in our model, but 0 is an unexpected value ==> it fails and displays comma in an infinite loop...

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