Y sequence is a difference sequence system introduced by a Japanese googologist Yukito.[1][2] It is intended to be an extension of hyper primitive sequence system, which is a also an extension of primitive sequence system. Although it has not been formalised yet, the expression \((1,3)\) in Y sequence is expected to correspond to the limit of Bashicu matrix system version 2.3, and its novel idea has given a new direction in googology.


An expression in Y sequence, which is also called a Y sequence, is a finite array \(a\) satisfying one of the following three conditions:

  1. \(a\) is the empty sequence \(()\).
  2. \(a\) is a non-empty sequence of positive numbers whose leftmost entry is \(1\).
  3. \(a\) is the sequence \((1,\omega)\).

For example, \((1)\), \((1,2,1,2)\), and \((1,3)\) are Y sequences, while \((0,1)\), \((2)\), or \((1,\omega+1)\) are not Y sequences.

We denote by \(\mathbb{Y}\) the set of Y sequences, and by \(\mathbb{N}_{>0}\) the set of positive integers. Assume that the expansion rule were formalised into a well-defined map \begin{eqnarray*} \textrm{Expand} \colon \mathbb{Y} \times \mathbb{N}_{>0} & \to & \mathbb{Y} \\ (a,n) & \mapsto & \textrm{Expand}(a,n). \end{eqnarray*} We define a partial computable function \begin{eqnarray*} \textrm{Y}[ \ ] \colon \mathbb{Y} \times \mathbb{N}_{>0} & \to & \mathbb{N} \\ (a,n) & \mapsto & \textrm{Y}a[n] \end{eqnarray*} in the following recursive way:

  1. If \(a = ()\), then \(\textrm{Y}a[n] = n\).
  2. If \(a\) is a non-empty sequence of positive integers whose leftmost entry is \(1\), then \(\textrm{Y}a[n] = \textrm{Y} \textrm{Expand}(a,n)[n]\).
  3. If \(a = (1,\omega)\), then \(\textrm{Y}a[n] = \textrm{Y}(1,n)[n]\).

Assume \(((1,\omega),n) \in \textrm{dom}([ \ ])\) for any \(n \in \mathbb{N}_{>0}\). Let \(f\) denote the total computable function \(n \mapsto \textrm{Y}(1,\omega)[n]\).

Yukito named \(f^{2000}(1)\) Y sequence number. Since \((1,3)\) is supposed to correspond to the limit of Bashicu matrix system version 2.3, Y sequence number is supposed to be significantly greater than Bashicu matrix number with respect to the version. We note that the well-definedness of Bashicu matrix number is unknown, and hence the statement might not make sense even if \(\textrm{Expand}\) will be fully defined.


Although Y sequence has not been formalised yet, Yukito has explained expansions for several examples. Here is a list of known expansions of Y sequences originally given by Yukito.

Y sequence \(a\) expansion \(\textrm{Expand}(a,-)\)
\((1)\) \(()\)
\((1,1)\) \((1)\)
\((1,1,1)\) \((1,1)\)
\((1,1,1,1)\) \((1,1,1)\)
\((1,2)\) \((1,1,1,1,\ldots)\)
\((1,2,1)\) \((1,2)\)
\((1,2,2)\) \((1,2,1,2,1,2,1,2,\ldots)\)
\((1,2,2,2)\) \((1,2,2,1,2,2,1,2,2,1,2,2,\ldots)\)
\((1,2,3)\) \((1,2,2,2,2,\ldots)\)
\((1,2,3,3)\) \((1,2,3,2,3,2,3,2,3,\ldots)\)
\((1,2,4)\) \((1,2,3,4,5,\ldots)\)
\((1,2,4,1)\) \((1,2,4)\)
\((1,2,4,2)\) \((1,2,4,1,2,4,1,2,4,1,2,4,\ldots)\)
\((1,2,4,3)\) \((1,2,4,2,4,2,4,2,4,\ldots)\)
\((1,2,4,4)\) \((1,2,4,3,5,4,6,5,7,\ldots)\)
\((1,2,4,5)\) \((1,2,4,4,4,4,\ldots)\)
\((1,2,4,6)\) \((1,2,4,5,7,8,10,11,13,\ldots)\)
\((1,2,4,7)\) \((1,2,4,6,8,10,\ldots)\)
\((1,2,4,8)\) \((1,2,4,7,11,16,\ldots)\)
\((1,3)\) \((1,2,4,8,\ldots)\)
\((1,3,1)\) \((1,3)\)
\((1,3,2)\) \((1,3,1,3,1,3,1,3,\ldots)\)
\((1,3,3)\) \((1,3,2,5,4,9,8,17,\ldots)\)
\((1,3,4)\) \((1,3,3,3,3,\ldots)\)
\((1,3,5)\) \((1,3,4,7,11,18,29,47,\ldots)\)
\((1,3,6)\) \((1,3,5,7,9,\ldots)\)
\((1,3,7)\) \((1,3,6,12,24,48,96,192,\ldots)\)
\((1,3,8)\) \((1,3,7,15,31,\ldots)\)
\((1,3,9)\) \((1,3,8,20,48,\ldots)\)
\((1,4)\) \((1,3,9,27,\ldots)\)
\((1,5)\) \((1,4,16,64,\ldots)\)
\((1,6)\) \((1,5,25,125,\ldots)\)
\((1,7)\) \((1,6,36,216,\ldots)\)

Needless to say, the table does not unqiuely characterise the expansion rule. Indeed, Yukito officially keeps the expansions of several Y sequences to be undecided.

Alternative Formalisations

Through the examples of expansions, several googologists are trying to find a formal rule (partially or essentially) consistent with the original expansions.[3][4][5] The formalisations by others are not regarded as an official defintion of Y sequence, and should be distinguished from the original Y sequence. Sometimes several googologists introduce their formalisation as "the definition of Y sequence" or something like that, it does not mean that they are allowed to express so by Yukito or even that those are compatible with the original explanation by Yukito.

Yukito stated that Y sequence is the name only for the difference sequence system which he himself will complete, and he did not want others to name their own difference sequence systems Y sequence version 1.1 or something like that. Therefore others tend to call their own difference sequence systems distinct names unless they directly ask Yukito permissions.


  1. The user page of Yukito in the Japanese Googology Wiki.
  2. Yukito, Y数列, Google Document.
  3. Hexirp, Y数列 Hexirp 版, Japanese Googology Wiki user blog post.
  4. p進大好きbot, New Difference Sequence System, Googology Wiki user blog post.
  5. Syst3ms, Bismuth : not Y sequence, but close, Googology Wiki user blog post.

See also

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Jason: Irrational arrow notation · δOCF · δφ
By mrna: SSAN · S-σ
By Nayuta Ito: N primitive
By Yukito: Hyper primitive sequence system · Y sequence · Y function
Indian counting system: Lakh · Crore · Uppala
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Other: Taro's multivariable Ackermann function · Sushi Kokuu Hen

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