Omega-Y sequence

\(\omega\)-Y sequence is a difference sequence system introduced by a Japanese googologist Yukito in July 2021.Yukito, twitter, July 17, 2021.Naruyoko, "Study and Expand Sequence(仮)", github io, July 17, 2021 It is intended to be much stronger than the creator's past work Y sequence.
\(\omega\)-Y sequence has new difference sequences called galaxy and galaxy group (or super cluster), which are the extensions of Mt. Fuji for the multidimensional structure. 
Y sequence has "diagonal difference sequence" which consists of the numbers picked up from the top of the Mt. Fuji and it make a new Mt.Fuji (1,4,24 on the top of the Mt.Fuji based on (1,5,33) in the figure). On the other hand, \(\omega\)-Y sequence has not only diagonal difference sequences but also galactic difference sequence(or spiral difference sequence), which consists of the numbers picked up at the bottom-left corners of Mt.Fujis (1,3,14 on the (1,5,33)-based galaxy). 
The galactic difference sequence (1,3,14) has an own difference sequence (2,11), and it makes a new galaxy based on the difference sequence (2,11) (in the right side of (1,5,33)). The galaxy difference sequence of the (2,1)-based galaxy is (2,7) and it makes another (5)-based galaxy. Now we got the 3 galaxies and it makes a new galaxy group based on (1,3) (in the bottom side of the (1,5,33)) by the difference sequence of the galaxy group difference sequence (1,2,5). 
In this way, \(\omega\)-Y sequence can continue it infinitely, sequences, Mt.Fujis, galaxies, galaxy groups, 6th structures, 7th structures, 8th structures, ... and so on. The n-th structures consist of n-th dimensional structure. Especially, \(\omega\)-Y(1,n) consist of n-th dimensional simplex.
The difference of the structure of \(\omega\)-Y sequence to the one of Y sequence is that Y sequence makes a next Mt.Fuji by putting the diagonal sequence itself into the base of the new Mt.Fuji and \(\omega\)-Y sequence makes the next Mt.Fuji (or the next structure) by putting the "difference sequence" of the diagonal sequence. For example, the basic Mt.Fuji of Y(1,4) is ((1,4),(3)) and the diagonal sequence is (1,3), and the next Mt.Fuji becomes (1,3) itself in Y-sequence. On the other hand in \(\omega\)-Y sequence, the diagonal sequence of ((1,4),(3)) is (1,3) and the next Mt.Fuji becomes (2) which is the difference sequence of (1,3). 
The first difference between Y sequence and \(\omega\)-Y sequence is the limit of (1,3,2,5,4,9,8,17,\(\cdots,2^k,2^{k+1}+1,\cdots\)), Y(1,3,3) on Y sequence and \(\omega\)-Y(1,3,2,5,5) on \(\omega\)-Y sequence.
Yukito saidYukito, twitter, Aug. 17, 2021 he had tried to make Y^n sequences whose limit Y^n(1,\(\omega\)) is equal to Y^n+1(1,3) at first time, however, it had critical bug and he changed the way. While Y^ns unlock the new dimensions, \(\omega\)-Y sequence unlocks all the dimensions at once.
==Definitions==
===Original definition===
Yukito declared the completion of the \(\omega\)-Y sequence and it was defined as the User:Naruyoko's program in twitterYukito, twitter, Sept. 1, 2021 
Yukito said that he coined \(\omega\)-Y sequence number using \(\omega\)-Y sequence as a masterpiece of Yukito by giving the detailed definition on a user blog laterYukito twitter, Sept. 1, 2021Yukito twitter, Sept. 1, 2021 also.
===Expansion rule===
The expansion rule of \(\omega\)-Y sequence to solve an expression like \(\omega\)-Y(s)[n] for a valid sequence \(s\) and a natural number \(n\) is defined by the function expand(s,n,stringify) on the code by Naruyoko.
===Large number===
Yukito said that he named \(f^{2000}(1)\) using f(n)=ω-Y\((1,\omega)[n]\) as \(\omega\)-Y sequence number.
==Programs==
User:Naruyoko made a program "Study and Expand Sequence(仮)" to expand \(\omega\)-Y sequences.
User:Naruyoko made a program MEGA whY mountain to draw the structure of \(\omega\)-sequence. It draws a multi-dimensional Mt. Fuji with a structure that looks like a pile of Mt.Fuji on top of Mt.Fuji, one after another. Yukito calls it Mega-Mt.Fuji.
== See also ==
* Y sequence
* YY sequence
== Sources ==
Category:Notations
Category:Numbers
Category:Higher computable level
Category:Difference sequence system
Category:Googology in Asia
Category:Sequences