[seqfan] Re: Long chains

wouter meeussen wouter.meeussen at pandora.be
Sun May 29 19:25:46 CEST 2011



contains the sentence (pg 21):
"From (1) and (2), we can have the summation form of orbn, but we
cannot find a simple formula. The sequence {orbn}1
n=0 starts with
1, 1, 2, 6, 18, 60, 210, 754, 2766, 10280, 38568, . . . , and it does
not appear in On-Line Encyclopedia of Integer Sequences."

and today, it still doesn't !
It would be quite challenging to explain its significance in plain language!
Quite beyond my length-of-stick


----- Original Message ----- 
From: "Charles Greathouse" <charles.greathouse at case.edu>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Friday, May 27, 2011 8:15 AM
Subject: [seqfan] Long chains

> 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 &
> 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
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