Definition:Earthworm Sequence

Definition

The earthworm sequence is the integer sequence $\sequence {h_n}$ defined as follows.

Let $h: \Z \to \Z$ be the mapping defined as:

$\forall a \in \Z: \map h a = 2 a \pmod {100}$

Then:

$h_n = \begin {cases} x: x \in \set {10, 11, \dotsc, 99} & : n = 0 \\ \map h {n - 1} & : n > 0 \end {cases}$

That is:

$h_0$ is an arbitrary integer between $10$ and $99$

$h_n$ is the result of multiplying the previous term in the sequence by $2$ and keeping only the last two digits.

The earthworm sequence is analysed for all $n \in \set {00, 01, 02, \dotsc, 99}$, not only for the range $\set {10, 11, 12, \dotsc, 99}$.

The first table lists the integers that end up in a loop of $20$ terms:

$n$	$\map h n$	$\map h {\map h n}$	$\map h {\map h {\map h n} }$
$01$, $51$	$02$	$04$	$08$
$27$, $77$	$54$	$08$	$16$
$29$, $79$	$58$	$16$	$32$
$33$, $83$	$66$	$32$	$64$
$41$, $91$	$82$	$64$	$28$
$07$, $57$	$14$	$28$	$56$
$39$, $89$	$78$	$56$	$12$
$03$, $53$	$06$	$12$	$24$
$31$, $81$	$62$	$24$	$48$
$37$, $87$	$74$	$48$	$96$
$49$, $99$	$98$	$96$	$92$
$23$, $73$	$46$	$92$	$84$
$21$, $71$	$42$	$84$	$68$
$17$, $67$	$34$	$68$	$36$
$09$, $59$	$18$	$36$	$72$
$43$, $93$	$86$	$72$	$44$
$11$, $61$	$22$	$44$	$88$
$47$, $97$	$94$	$88$	$76$
$19$, $69$	$38$	$76$	$52$
$13$, $63$	$26$	$52$	$04$

The second table lists the integers that end up in a loop of $4$ terms:

$n$	$\map h n$	$\map h {\map h n}$	$\map h {\map h {\map h n} }$
$05$, $55$	$10$	$20$	$40$
$35$, $85$	$70$	$40$	$80$
$45$, $95$	$90$	$80$	$60$
$15$, $65$	$30$	$60$	$20$

The second table lists the integers that end up in a loop of $1$ term, that is, the degenerate case that ends in $0$:

$25$, $75$

$50$

$00$

His interest is specifically in whether each of the integers selected for $h_1$ ever returns to its starting point.

He fails to document the details of what happens for $h_0 = 50$.

1991: Clifford A. Pickover: Computers and the Imagination: Part $\text V$. Exploration: Chapter $41$. Earthworm Algebra