[seqfan] Continued Fractions

[Melvin M Peralta]

Hello SeqFans,

a(0) = 1
a(n+1) = numerator of the simplified continued fraction resulting from
[a(0), a(1), ..., a(n)]

1, 1, 2, 5, 27, 734, 538783, ...

a(n) is also the number of ways to tile an n-board with dominos and
stackable squares, where nothing can be stacked on a domino but
otherwise the i-th cell may be stacked by as many as a_i squares. (a nice
proof of this is given in "Proofs That Really Count").

*My question*: From n>=2, the sequence appears identical to
https://oeis.org/A058182. Does it always coincide? If so is the connection
already implicit in the entry? Or should a comment be made?

Note a similar sequence does not appear in OEIS:
a(0) = 1
a(n+1) = denominator of the simplified continued fraction resulting from
[a(0), a(1), ..., a(n)]

1, 1, 2, 3, 10, 103, 10619, ...

This is also nice because a(n+1) is the number of ways to tile an n-board
in the same way described, except the first cell should be ignored entirely.

Best Regards,
Melvin