[seqfan] Long chains

[Charles Greathouse]

I was looking at a sequence, A179771, trying to check it and a
conjecture in its comment.  Its definition is
a(n) = A179770(4*n)
so I looked up that sequence, defined as
The fourth central column of triangle A122242, i.e. A179761(4),
A179761(11), A179761(20), A179761(31), ...
and so I searched for A179761, defined as
Binary expansions of A122242 (A122243) concatenated together to a
single binary sequence, so that from each term of A122242, the most
significant bits come before the least significant bits.
where A1222243 is defined as
a(n) = A007088(A122242(n))
the exterior being the binary representation and the interior being
a(n) = A014486(A122241(n)).

A014486 is
List of totally balanced sequences of 2n binary digits written in base
10. Binary expansion of each term contains n 0's and n 1's and reading
from left to right (the most significant to the least significant
bit), the number of 0's never exceeds the number of 1's.
and
A122241 is
Iterates of A122237, starting from 4.
where
A122237 is
a(n) = A057548(A082358(n))
where
A057548 is
A014486-indices of Catalan mountain ranges with no sea-level valleys,
i.e. the rooted plane general trees with root degree = 1.
and A014486 is as above, and
A082358 is
Permutation of natural numbers: composition of permutations A057163 & A082356.

OK, so now I just need to understand those sequences.  A057163 is
Signature-permutation of a Catalan automorphism: Reflect a rooted
plane binary tree; Deutsch's 1998 involution on Dyck paths.
which does not enlighten me, but since it doesn't mention any other
sequences I'll just follow the other branch: A082356 is
Permutation of natural numbers induced by the gatomorphism gma082356
acting on the parenthesizations encoded by A014486/A063171.
which relies on the custom definition of gma082356, the definition of
A014486 above, and A063171,
Dyck language interpreted as binary numbers in ascending order.

It may be that I am simply an inferior mathematician, but I can't hold
all that in my head, let alone judge what meaning it might have or
whether it is significant.  Worse, it's not even clear whether it's
well-defined: can anyone even tell if the directed definition graph is
acyclic without charting it out?  (In case you were lost, in my
preceding paragraphs I made two backreferences.)

This may be an isolated example, but please treat this as a plea to
(1) define sequences in words rather than A-numbers when possible and
reasonable, and (2) explain the significance of the sequence.  Perhaps
some great mind can look at this landscape and understand how picking
a column from a triangle based on the base-2 and -10 representations
of numbers based on an iteration of a sequence based on Catalan
numbers and a particular permutation related to Dyck paths and the
gatomorphism gma082356 'makes sense'... but I can't, and I suspect
most of the readers of the OEIS can't.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University