# Preparation

Paradigm(s) String-rewriting User:Hakerh400 2023 Unknown Implemented .txt

Preparation is an esolang invented by User:Hakerh400 in 2023.

## Syntax

Source code consists of a single natural number.

## Overview

Let ${\displaystyle z}$ be the natural number from the source code. We define infinite sequence ${\displaystyle a_{i}}$ such that ${\displaystyle a_{0}=z}$ and ${\displaystyle a_{i+1}=k_{i}\left(k_{i}^{2}-a_{i}\right)}$ for any natural number ${\displaystyle i}$, where ${\displaystyle k_{i}=\left\lceil {\sqrt {a_{i}+1}}\right\rceil }$. Notation ${\displaystyle \left\lceil x\right\rceil }$ represents the ceiling function.

## Input/Output

There is no input. The output is the sequence ${\displaystyle a_{i}}$ up to the first repeating number.

## Example

Let the source code be ${\displaystyle 1234}$. Here is the sequence ${\displaystyle a_{i}}$:

1234
2232
3456
1475
1794
2365
1764
3655
4026
4480
603
550
624
25
66
135
108
143
12
16
45
28 <--\
48    |
7     |
6     |
9     |
28 ---/


Explanation. Start with ${\displaystyle 1234}$. Calculate ${\displaystyle k_{0}=\left\lceil {\sqrt {1234+1}}\right\rceil =36}$ and ${\displaystyle a_{1}=36\cdot (36^{2}-1234)=2232}$. Continue similarly for the rest of the sequence.

## Computational class

The author conjectures that this language is total.