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\(\omega\)-Y sequence is a difference sequence system introduced by a Japanese googologist Yukito in July 2021.[1][2] It is intended to be much stronger than the creator's past work Y sequence, as we will explain later.

\(\omega\)-Y sequence has new difference sequences called galaxy, galaxy group(or super cluster) and above, which are the extension of Mt. Fuji for the multidimensional structure.

Structure of ω-Y(1,5,33)

Structure of ω-Y(1,5,33)

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 (1,3,9,27,81,\(\cdots, 3^k, \cdots\)), Y(1,4) on Y sequence and \(\omega\)-Y(1,3,10) on \(\omega\)-Y sequence.

Yukito said[3] 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 the all dimensions at once.


Original definition

Yukito declared the completion of the \(\omega\)-Y sequence and it was defined as the User:Naruyoko's program in twitter[4]

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 later[5][6] 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)=\(\omega\)-Y(1,\(\omega\))[n] as \(\omega\)-Y sequence number[5].


Stacked drawing of ω-Y(1,5,33)

Stacked drawing of ω-Y(1,5,33)

User:Naruyoko made a program "Study and Expand Sequence(仮)"[2] 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

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's \(\psi\)
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 computation programmes · TR function (I0 function)
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence · YY sequence · Y function · ω-Y sequence
See also: Template:Googology in Asia