Definition:Pea Pattern/Ascending
Definition
The ascending pea pattern is defined as follows:
The first term $p_0$ is an arbitrary integer.
$p_n$ is formed from $p_{n - 1}$ as follows.
First, the distinct digits in $p_{n - 1}$ are counted, and the number of each is noted.
The count of each distinct digit is concatenated with an instance of the digit itself.
Then those concatenations are themselves concatenated into $p_n$ according to a predetermined order.
For the ascending pea pattern, the concatenation into $p_n$ is in ascending order of the distinct digits of $p_{n - 1}$.
Examples
Ascending Pea Pattern on $111$
| \(\ds \) | \(\) | \(\ds 111\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31\) | three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 1113\) | one $1$, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 3113\) | three $1$s, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 2123\) | two $1$s, two $3$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 112213\) | one $1$, two $2$s, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 312213\) | three $1$s, two $2$s, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 212223\) | two $1$s, two $2$s, two $3$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 114213\) | one $1$, four $2$s, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31121314\) | three $1$s, one $2$, one $3$, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 41122314\) | four $1$s, one $2$, two $3$s, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31221324\) | three $1$s, two $2$s, one $3$, two $4$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 21322314\) | two $1$s, three $2$s, two $3$s, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 21322314\) | two $1$s, three $2$s, two $3$s, one $4$ |
and a fixed point has been reached.
Ascending Pea Pattern on $231$
| \(\ds \) | \(\) | \(\ds 231\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 111213\) | one $1$, one $2$, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 411213\) | four $1$s, one $2$, one $3$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31121314\) | three $1$s, one $2$, one $3$, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 41122314\) | four $1$s, one $2$, two $3$s, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31221324\) | three $1$s, two $2$s, one $3$, two $4$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 21322314\) | two $1$s, three $2$s, two $3$s, one $4$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 21322314\) | two $1$s, three $2$s, two $3$s, one $4$ |
and a fixed point has been reached.
Ascending Pea Pattern on $10,213,223$
| \(\ds \) | \(\) | \(\ds 10213223\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 10213223\) | one $0$, two $1$s, three $2$s, two $3$s |
and so is seen to be a fixed point.
Ascending Pea Pattern on $1,031,223,314$
| \(\ds \) | \(\) | \(\ds 1031223314\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 1031223314\) | one $0$, three $1$s, two $2$s, three $3$s, one $4$ |
and so is seen to be a fixed point.
Linguistic Note
The origin of the term pea pattern is unclear.
The terms ascending pea pattern, descending pea pattern and sequential pea pattern appear not to be standard, as many sources focus on one of the variants and ignore the others.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10,213,223$