OH HO HO - Merry Chrstmas

[Carl G.]

An elf at the North Pole had a little too much "holiday cheer" and, after
noticing that "HO" is "OH" both upside down and backwards, came up with the
following problem:

Begin with the word "OH". Starting with the second letter ("H") append a
"HO" for each "H" and an "OH" for each "O", letting the message grow as
words are added. The first few steps are given below:

1. OH Start with "OH, which starts with
"O".
2. OH HO Added "HO" since the second letter is
"H"
3. OH HO HO Added "HO" since the third letter is "H"
4. OH HO HO OH Added "OH" since the fourth letter is "O"
5. OH HO HO OH HO Added "HO" since the fifth letter is "H"
6. OH HO HO OH HO OH Added "OH" since the sixth letter is "O"

When the sequence is built in this manner, the first letter of each word
will give the sequence itself (ignoring spaces).

Question:

Will the sequence ever begin repeating itself (like a repeating decimal)?
If so, after how many letters? If not, give a proof.

Carl G.

[Ed Murphy]

[spoiler space]





















Pretty sure this is equivalent to http://oeis.org/A010060 - thus there
is no one substring that gets infinitely repeated.