東巨3エントリー

[1]

L-階差数列類

表記

• \(a_0,a_1,...a_k:\)非負整数
• \(n:\)自然数
• \(M:\)L関数

• 正規形\(:(a_0,a_1,...a_k)_{M}[n]=(S)_M[n]\)

定義

サブルール

• \(c,e,g:\)自然数

• 列の長さ

列\(s\)の長さを\(|s|\)と表記する。

例:\(|0,1,1,2,2,3|=6\)

• 結合

列\(s\)と列\(t\)の結合を\(s\frown t\)と表記する。

例:\(0,1,1,2\frown 2,3=0,1,1,2,2,3\)

• 展開度

列\(s=b_0,b_1,...b_c\)の「展開度」を\(max\{e|∃s'[∃g[s=s'+g×0\frown s'+g×1\frown ...s'+g×e]]\}\)と定義する。

例:\(s=0,1,1,2,2,3,3,4\)

\(s'=0,1,1,2→s=s'+0×2\frown S'+1×2→e=1\)

\(s'=0,1→s=s'+0×1\frown s'+1×1\frown s'+2×1\frown s'+3×1→e=3\)

• L関数

L関数は、非負整数\(m\)と再帰順序数\({\alpha}\)を用いて\(L_{\alpha}(m)\)と表記され、列\(S\)に依存した関数である。

rule1:\(L_0(m)=\)「1以上の展開度を持ち、階差数列が全て-mより大きいようなSの連続部分列の最大長」

rule2:\(L_{\alpha+1}(m)=L^m_{\alpha}(m)\)

rule3:\(L_{\alpha}(m)=L_{\alpha[m]}(m):\)

• \(a_l\)

\(M\)の値に対応する列の内、最も左側にある列の初項を\(a_l\)とする。

• \(d_0,d_1,...d_k\)

\(d_0,d_1,...d_k\)を以下の通り定義する。

rule1:\(d_0=-\infty\)

rule2:\(d_k=max\{a_k-a_{k-1},-M\}\)

rule3:\(d_l=min\{a_k-a_{k-1},-M-1\}\)

rule4:\(d_c=a_c-a_{c-1}\)

• \(\text{ex}(S,b)\)

列\(S\)と自然数\(b\)から列への写像\(\text{ex}(s,n)\)を以下の通り定義する。

• \(r=max\{i|(a_i<a_k)∧(d_i<d_k)\}\)
• \(A=a_0,a_1,...a_{r-1}\)、\(B_0=a_r,a_{r+1},...a_{k-1}\)
• \(B_c=B_0+c×(a_k-a_r-1)\)
• \(\text{ex}(S,n)=A\frown B_0\frown B_1\frown ...B_n\)

計算法

• \(\#:\)長さ1以上の列

rule1:\((0)_M[n]=n+1\)

rule2:\((\#,0)_M[n]=(\#)_M[n+1]\)

rule3:\((\#)_M[n]=(\text{ex}(\#,n))_M[n]\)

命名

• \(A_1(x)=(0,2)_{L_0(0)}[x]\)
• \(A_2(x)=(0,2)_{L_0(1)}[x]\)
• \(A_3(x)=(0,2)_{L_0(L_0(0))}[x]\)
• \(A_4(x)=(0,2)_{L_{\omega}(x)}[x]\)
• \(A_5(x)=(0,2)_{L_{\varphi({\omega},0)}(x)}[x]\)

• E3:A-01-Hs\(=A^{108}_1(108)\)
• E3:A-02-Hs\(=A^{108}_2(108)\)
• E3:A-03-Hs\(=A^{108}_3(108)\)
• E3:A-04-Hs\(=A^{108}_4(108)\)
• E3:A-05-Hs\(=A^{108}_5(108)\)

計算例

• \((0,1,2,1,2,2,3,3,4,0,1,1,2,2,3,2)_{L_0(1)}[2]\)
  • \(L_0(1)=6\)
  • \(l=3\)
  • \(d=-\infty,1,1,-7,1,0,1,0,1,-4,1,0,1,0,1,-1\)
  • \(r=9\)
  • \(A=0,1,2,1,2,2,3,3,4\)、\(B_0=0,1,1,2,2,3\)
  • \(a_k-a_r-1=1\)

\((0,1,2,1,2,2,3,3,4,0,1,1,2,2,3,2)_{L_0(1)}[2]\)

\(=(0,1,2,1,2,2,3,3,4,0,1,1,2,2,3,1,2,2,3,3,4,2,3,3,4,4,5)_{L_0(1)}[2]\)

評価

\(S_0=0\)

\(S_{n+1}=S_n,S_n+1\)

• \(L_0(0)\)

\((0,1,2)=(|0,1|1,2|2,3|...)\)

\((0,1,1,2,2)=(|0,1,1,2|1,2,2,3|2,3,3,4|...)\)

\((0,1,1,2,1,2,2,3,2)=(|0,1,1,2,1,2,2,3|1,2,2,3,2,3,3,4|2,3,3,4,3,4,4,5|...)\)

\((0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,2)=(S_4,2)=(|S_4|S_4+1|S_4+2|...)\)

\((S_5,2)=(S_4|S_4+1|S_4+1|S_4+1|...)\)

\((S_4,2)={\varepsilon_0}\)

\((S_4,2,S_4,2)=(|S_4,2,S_4|S_4+1,3,S_4+1|S_4+2,4,S_4+2|...)={\varepsilon_1}\)

\((S_4,2,1)={\psi_0({\omega})}\)

\((S_4,2,1,0,1,1,2,S_4,2)={\psi_0({\omega+1})}\)

\((S_4,2,1,0,1,1,2,S_4,2,1)={\psi_0({\omega×2})}\)

\((S_4,2,1,0,1,1,2,1)={\psi_0({\omega^2})}\)

\((S_4,2,1,S_4,2)={\psi_0({\Omega})}\)

\((S_4,2,1,S_4,2,S_4,2,1,S_4,2)={\psi_0({\Omega×2})}\)

\((S_4,2,1,S_4,2,1)={\psi_0({\Omega×{\omega}})}\)

\((S_4,2,1,S_4,2,1,0,1,1,2,S_4,2,1,S_4,2)={\psi_0({\Omega×{\omega+1}})}\)

\((S_4,2,1,S_4,2,1,S_4,2)={\psi_0({\Omega^2})}\)

\((S_4,2,1,1)={\psi_0({\Omega^{\omega}})}\)

\((S_4,2,1,1,0,1,1,2,S_4,2,1,1)={\psi_0({\Omega^{\omega}×2})}\)

\((S_4,2,1,1,0,1,1,2,1)={\psi_0({\Omega^{\omega}×{\omega}})}\)

\((S_4,2,1,1,0,1,1,2,1,0,1,1,2,1)={\psi_0({\Omega^{\omega}×{\omega^2}})}\)

\((S_4,2,1,1,S_4,2)={\psi_0({\Omega^{\omega+1}})}\)

\((S_4,2,1,1,S_4,2,1)={\psi_0({\Omega^{\omega+1}×{\omega}})}\)

\((S_4,2,1,1,S_4,2,1,S_4,2)={\psi_0({\Omega^{\omega+2}})}\)

\((S_4,2,1,1,1)={\psi_0({\Omega^{\omega^2}})}\)

\((S_4,2,1,2)={\psi_0({\Omega^{\omega^{\omega}}})}\)

\((S_4,2,S_3+1,2)={\psi_0({\Omega^{\Omega}})}\)

\((S_4,2,S_3+1,2,1)={\psi_0({\Omega^{\Omega}×{\omega}})}\)

\((S_4,2,S_3+1,2,1,S_4,2)={\psi_0({\Omega^{\Omega+1}})}\)

\((S_4,2,S_3+1,2,1,1)={\psi_0({\Omega^{\Omega×{\omega}}})}\)

\((S_4,2,S_3+1,2,S_3,2)={\psi_0({\Omega^{\Omega^2}})}\)

\((S_4,2,2)={\psi_0({\Omega^{\Omega^{\omega}}})}\)

\((S_4,S_3+2,3)=(S_3,S_3+1,S_3+2,3)={\psi_0({\Omega^{\Omega^{\Omega}}})}\)

\((S_3,2)={\psi_0({\psi_1(0)})}\)

\((S_3,2,1)={\psi_0({\psi_1(0)×{\omega}})}\)

\((S_3,2,2)=(S_2|S_2+1,2|S_2+1,2|S_2+1,2|...)\)

\((S_3,2,1,0,1,1,2,S_3,2)={\psi_0({\psi_1(0)×({\omega+1})})}\)

\((S_3,2,1,0,1,1,2,1)={\psi_0({\psi_1(0)×{\omega^2}})}\)

\((S_3,2,1,S_4,2)={\psi_0({\psi_1(0)×{\Omega}})}\)

\((S_3,2,1,S_3,2)=(|S_3,2,1,S_3|S_3+1,3,2,S_3+1|S_3+2,4,3,S_3+2|...)\)

\((S_3,2,1,S_3,S_3+1,3)=(S_3,2,1,S_3|S_3+1|S_3+2|S_3+3|...)\)

\((S_3,2,1,S_4,2,1,S_4,2)={\psi_0({\psi_1(0)×{\Omega^2}})}\)

\((S_3,2,1,S_4,2,1,1)={\psi_0({\psi_1(0)×{\Omega^{\omega}}})}\)

\((S_3,2,1,S_4,2,S_3+1,2)={\psi_0({\psi_1(0)×{\Omega^{\Omega}}})}\)

\((S_3,2,1,S_3,S_3+1,3)={\psi_0({\psi_1(0)^2})}\)

\((S_3,2,1,S_3,2)={\psi_0({\psi_1(1)})}\)

\((S_3,2,1,1)={\psi_0({\psi_1({\omega})})}\)

\((S_3,2,1,1,S_3,2,1,1)={\psi_0({\psi_1({\omega^2})})}\)

\((S_3,2,1,2)={\psi_0({\psi_1({\omega^{\omega}})})}\)

\((S_3,2,2)=(S_2|S_2+1,2|S_2+1,2|S_2+1|...)\)

\((S_3,2,S_2+1,2)=(|S_3,2,S_2+1|S_3+1,3,S_2+2|S_3+2,4,S_2+3|...)\)

\((S_3,2,S_2+1,S_3+1,3)=(S_3,2,S_2+1|S_3+1|S_3+2|S_3+3|...)\)

\((S_3,2,S_2+1,S_4+1,2)={\psi_0({\psi_1({\Omega})})}\)

\((S_3,2,S_2+1,S_3+1,3)={\psi_0({\psi_1({\psi_1(0)})})}\)

\((S_3,2,S_2+1,2)={\psi_0({\psi_1({\Omega_2})})}\)

\((S_3,2,2)={\psi_0({\psi_1({\Omega_2^{\omega}})})}\)

\((S_2,S_2+1,S_2+2,3)={\psi_0({\psi_1({\Omega_2^{\Omega_2}})})}\)

\((S_2,2)=(0,1,1,2,2)={\psi_0({\psi_1({\psi_2(0)})})}\)

\((S_2,2,1)={\psi_0({\psi_1({\psi_2(0)})}×{\omega})}\)

\((S_2,2,1,S_2,2)={\psi_0({\psi_1({\psi_2(0)})}×({\omega+1}))}\)

\((S_2,2,2)=(|S_2,2|S_2+1,3|S_2+2,4|...)\)

\((S_2,2,S_2+1,3)=(S_2,2|S_2+1|S_2+2|S_2+3|...)\)

\((S_2,2,S_4+1,3)={\psi_0({\psi_1({\psi_2(0)})}×{\Omega})}\)

\((S_2,2,S_3+1,3)={\psi_0({\psi_1({\psi_2(0)}+1)})}\)

\((S_2,2,S_2+1,3)={\psi_0({\psi_1({\psi_2(0)×2})})}\)

\((S_2,2,2)={\psi_0({\psi_1({\psi_2(1)})})}\)

\((S_2,2,3)={\psi_0({\psi_1({\psi_2({\omega})})})}\)

\((S_2,2,3,3)=(|S_2,2,3|S_2+2,4,5|S_2+4,6,7|...)\)

\((S_2,2,3,S_4+2,4)={\psi_0({\psi_1({\psi_2({\Omega})})})}\)

\((S_2,2,3,S_3+2,4)={\psi_0({\psi_1({\psi_2({\Omega_2})})})}\)

\((S_2,2,3,S_2+2,4)={\psi_0({\psi_1({\psi_2({\psi_2(0)})})})}\)

\((0,1,1,2,2,3,3)={\psi_0({\psi_1({\psi_2({\Omega_3})})})}\)

\((0,1,2)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})})}\)

\((0,1,2,S_3,S_5,2)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+1)}\)

\((0,1,2,S_3,S_5,2,S_3,1)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+{\omega})}\)

\((0,1,2,S_3,S_5,2,S_3,1,S_3,S_5,2)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+{\omega+1})}\)

\((0,1,2,S_3,S_5,2,S_3,1,S_3,1)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+{\omega^2})}\)

\((0,1,2,S_3,S_5,2,S_3,1,1)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+{\omega^{\omega}})}\)

\((0,1,2,S_3,S_5,2,S_3,S_5+1,3)={\psi_0({\psi_1({\psi_2({\Omega_3^{\omega}})})}+{\Omega})}\)

\((0,2)={\psi_0({\Omega_{\omega}})}\)

• \(L_0(1)\)

\((S_6,2)={\varepsilon_0}\)

\((S_6,2,S_4,S_6,2)={\varepsilon_1}\)

\((S_6,2,S_4,1)={\psi_0({\omega})}\)

\((S_6,2,S_6,2)=(|S_6,2,S_6|S_6+1,3,S_6+1|S_6+2,4,S_6+2|...)\)

\((S_6,2,S_4,1,S_4,S_6,2)={\psi_0({\omega+1})}\)

\((S_6,2,S_4,1,S_4,1)={\psi_0({\omega^2})}\)

\((S_6,2,S_4,1,1)={\psi_0({\omega^{\omega}})}\)

\((S_6,2,S_6,2)={\psi_0({\Omega})}\)

\((S_6,2,S_6,2,S_4,S_6,2,S_6,2)={\psi_0({\Omega×2})}\)

\((S_6,2,S_6,2,S_6,2)={\psi_0({\Omega^2})}\)

\((S_6,2,1)={\psi_0({\Omega^{\omega}})}\)

\((0,1,2)={\psi_0({\Omega_{\omega}})}\)

\((0,1,2,1)={\psi_0}({\Omega_{\omega}})×{\omega}\)

\((0,1,2,1,2,2,3)={\psi_0({\Omega_{\omega}})×{\omega^{\omega^{\omega}}}}\)

\((0,1,2,1,2,2,3,0,1,2,S_3,1)={\psi_0({\Omega_{\omega}})}×({\omega^{\omega^{\omega}}+1})\)

\(L_{\Omega}-\)階差数列システム

表記

L-階差数列類を参照。

定義

サブルール

• L関数

順序数の代わりに数列を使う。

rule0-0:\((0)_M=1\)

rule0-1:\((\#,0)_M=(\#)_M+1\)

rule0-2:\((\#)_M[n]=(\text{ex}(\#,n))_M\)

計算法

L-階差数列類を参照。

命名

• \(f(x)=(0,2)_{L_x(x)}\)
• \(A_6(x)=(0,2)_{L_{f^x(x)}(x)}[x]\)

• E3:A-06-Hs\(=A^{108}_6(108)\)

大偽行列システム

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...a_{x,m})\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

計算法

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=max\{b|a_{x,b}>0\}\)
• \(p_0(x)=max\{b|(a_{b,0}+1=a_{x,0})∧(b<x)\}\)
• \(p_{y+1}(x)=max\{b|(a_{b,y+1}+1=a_{x,y+1})∧(b<x)∧(∃c[p_y^c(x)=b])\}\)
• \(r=max\{b|(∀y,y≤dim(k)[∃c[p_y^c(k)=b]])∧\)

\(((p_{dim(k)}(b)≠b-1)∨(∃e[(e>dim(k))∧(p_e(b)≠b-1)∧(a_{b,e}>0)]))\}\)

• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,y}-a_{r,y}-1&\text{if}　y=dim(k)\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[p_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

命名

• \(B_1(x)=(0,0,0)(1,1,1)[x]\)
• \(B_2(x)=(\underbrace{0,0,...0}_x)(\underbrace{1,1,...1}_x)[x]\)
• E3:B-01-Hs\(=B_1^{108}(108)\)
• E3:B-02-Hs\(=B_2^{108}(108)\)

計算例

• \((0,0,0,0)(1,1,1,1)(2,2,2,1)(3,3,3,0)(4,4,4,0)(5,0,0,0)(4,4,4,0)[1]\)

• \(dim(6)=2\)
• \(p_0(6)=3,p_0^2(6)=2,p_0^3(6)=1,p_0^4(6)=0\)
• \(p_1(6)=3,p_1^2(6)=2,p_1^3(6)=1,p_1^4(6)=0\)
• \(p_2(6)=3,p_2^2(6)=2,p_2^3(6)=1,p_2^4(6)=0\)
• \(r=2　(e=3)\)
• \(D_0=(2,2,1,0)\)
• \({\Delta}=(2,2,1,0)(2,2,1,0)(2,2,1,0)(2,0,0,0)\)
• \(A=(0,0,0,0)(1,1,1,1)\)
• \(B_0=(2,2,2,1)(3,3,3,0)(4,4,4,0)(5,0,0,0)\)
• \(B_1=(4,4,3,1)(5,5,4,0)(6,6,5,0)(7,0,0,0)\)

\((0,0,0,0)(1,1,1,1)(2,2,2,1)(3,3,3,0)(4,4,4,0)(5,0,0,0)(4,4,4,0)[1]\)

\(=(0,0,0,0)(1,1,1,1)(2,2,2,1)(3,3,3,0)(4,4,4,0)(5,0,0,0)(4,4,3,1)(5,5,4,0)(6,6,5,0)(7,0,0,0)[1]\)

評価

\((0,0)(1,1)=|(0,0)|(1,0)|(2,0)|...={\psi_0({\Omega^{\omega}})}\)

\((0,0)(1,1)(2,0)=|(0,0)(1,1)|(1,0)(2,1)|(2,0)(3,1)|...={\psi_0({\Omega^{\omega}+1})}\)

\((0,0)(1,1)(2,0)(3,0)={\psi_0({\Omega^{\omega}+{\Omega}})}\)

\((0,0)(1,1)(2,0)(3,1)=(0,0)(1,1)|(2,0)|(3,0)|(4,0)|...={\psi_0({\Omega^{\omega}×2})}\)

\((0,0)(1,1)(2,0)(3,1)(4,0)(5,1)={\psi_0({\Omega^{\omega}×3})}\)

\((0,0)(1,1)(2,1)=|(0,0)(1,1)|(2,0)(3,1)|(4,0)(5,1)|...={\psi_0({\Omega^{\omega}×{\omega}})}\)

\((0,0)(1,1)(2,1)(3,0)=(0,0)(1,1)|(2,1)|(2,1)|(2,1)|...\)

\((0,0)(1,1)(2,1)(2,1)=|(0,0)(1,1)(2,1)|(2,0)(3,1)(4,1)|(4,0)(5,1)(6,1)|...\)

\((0,0)(1,1)(2,1)(2,0)(3,1)(4,1)={\psi_0({\Omega^{\omega}×{\omega}+{\psi_0({\Omega^{\omega}×{\omega}})}})}\)

\((0,0)(1,1)(2,1)(2,1)={\psi_0({\Omega^{\omega}×{\omega}+{\Omega}})}\)

\((0,0)(1,1)(2,1)(3,0)={\psi_0({\Omega^{\omega}×{\omega}+{\Omega×{\omega}}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^2}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(2,1)(3,0)(3,1)={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^2×2}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(3,1)={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^3}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(4,0)={\psi_0({\Omega^{\omega}×({\omega+1})})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(4,0)(3,1)={\psi_0({\Omega^{\omega+1}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(4,0)(4,1)={\psi_0({\Omega^{\Omega}})}\)

\((0,0)(1,1)(2,1)(3,0)(3,1)(4,0)(4,1)(5,0)(5,1)={\psi_0({\Omega^{\Omega^{\Omega}}})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)=(0,0)(1,1)|(2,1)(3,0)|(3,1)(4,0)|(4,1)(5,0)|...={\psi_0({\psi_1(0)})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,0)=(0,0)(1,1)|(2,1)(3,0)(4,0)|(3,1)(4,0)(5,0)|(4,1)(5,0)(6,0)|...\)

\(={\psi_0({\psi_1(1)})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,0)(5,0)={\psi_0({\psi_1({\omega})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,1)=(0,0)(1,1)(2,1)|(3,0)(4,0)|(4,0)(5,0)|(5,0)(6,0)|...\)

\(={\psi_0({\psi_1({\varepsilon_0})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)=(0,0)(1,1)|(2,1)(3,0)(4,0)|(4,1)(5,0)(6,0)|(6,1)(7,0)(8,0)|...\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,1)(4,1)={\psi_0({\psi_1({\varepsilon_1})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,1)(5,0)(5,1)={\psi_0({\psi_1({\psi_0({\Omega})})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(4,1)(5,0)(6,0)={\psi_0({\psi_1({\psi_0({\psi_1(0)})})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)={\psi_0({\psi_1({\Omega})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(2,1)(3,0)(4,0)(5,0)={\psi_0({\psi_1({\Omega})}+{\psi_1({\psi_0({\psi_1({\Omega})})})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(3,0)={\psi_0({\psi_1({\Omega})}+{\psi_1({\psi_0({\psi_1({\Omega})})})}×{\omega})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(4,0)={\psi_0({\psi_1({\Omega})}+{\psi_1({\psi_0({\psi_1({\Omega})})}+1)})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(4,1)(5,0)(6,0)(7,0)={\psi_0({\psi_1({\Omega})}+{\psi_1({\psi_0({\psi_1({\Omega})×2})})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(5,0)={\psi_0({\psi_1({\Omega})×2})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(5,1)={\psi_0({\psi_1({\Omega})}×{\varepsilon_0})}\)

\((0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(6,0)={\psi_0({\psi_1({\Omega})×{\Omega}})}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)={\psi_0({\psi_1({\Omega})×{\Omega^{\omega}}})}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)(2,1)(3,0)(4,1)={\psi_0({\psi_1({\Omega})×{\Omega^{\omega}}}+{\psi_1({\psi_0({\psi_1({\Omega})×{\Omega^{\omega}}})})})}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)(4,0)={\psi_0({\psi_1({\Omega})×{\Omega^{\omega}}}+{\psi_1({\psi_0({\psi_1({\Omega})×{\Omega^{\omega}}})}+1)})}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)(5,0)={\psi_0({\psi_1({\Omega})}×({\Omega^{\omega}+1}))}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)(5,0)(6,1)={\psi_0({\psi_1({\Omega})}×{\Omega^{\omega}×2})}\)

\((0,0)(1,1)(2,1)(3,0)(4,1)(5,1)={\psi_0({\psi_1({\Omega})}×{\Omega^{\omega}}×{\omega})}\)

\(\)

以下、めっちゃ雑な予想↓

\((0,0)(1,1)(2,2)=|(0,0)(1,1)|(2,1)(3,2)|(4,2)(5,3)|...={\psi_0({\Omega_{\omega}})}=B(0,0,0)(1,1,1)\)

\((0,0)(1,1)(2,2)(2,0)={\psi_0({\Omega_{\omega}+1})}=B(0,0,0)(1,1,1)(1,1,0)\)

\((0,0)(1,1)(2,2)(3,0)={\psi_0({\Omega_{\omega}+{\Omega}})}=B(0,0,0)(1,1,1)(1,1,0)(2,1,0)\)

\((0,0)(1,1)(2,2)(3,0)(4,1)={\psi_0({\Omega_{\omega}+{\Omega^{\omega}}})}=B(0,0,0)(1,1,1)(1,1,0)(2,1,0)(3,0,0)\)

\((0,0)(1,1)(2,2)(3,0)(4,1)(5,2)={\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}=B(0,0,0)(1,1,1)(1,1,0)(2,2,1)\)

\((0,0)(1,1)(2,2)(3,1)={\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})×{\omega}}})}\)

\((0,0)(1,1)(2,2)(3,1)(3,0)={\psi_0({\Omega_{\omega}}+{\psi_1({\Omega_{\omega}})}+{\Omega})}\)

\((0,0)(1,1)(2,2)(3,1)(3,1)={\psi_0({\Omega_{\omega}}+{\psi_1({\Omega_{\omega}})}+{\Omega^2})}\)

\((0,0)(1,1)(2,2)(3,1)(4,0)(5,0)={\psi_0({\Omega_{\omega}}+{\psi_1({\Omega_{\omega}})}+{\psi_1(0)})}\)

\((0,0)(1,1)(2,2)(3,2)={\psi_0({\Omega_{\omega}×2})}=B(0,0,0)(1,1,1)(1,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(3,2)={\psi_0({\Omega_{\omega}×3})}=B(0,0,0)(1,1,1)(1,1,1)(1,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)={\psi_0({\Omega_{\omega}×{\omega}})}=B(0,0,0)(1,1,1)(2,0,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(3,2)={\psi_0({\Omega_{\omega}×({\omega+1})})}\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(4,1)={\psi_0({\Omega_{\omega}×{\Omega}})}=B(0,0,0)(1,1,1)(2,1,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(4,1)(3,2)={\psi_0({\Omega_{\omega}^2})}=B(0,0,0)(1,1,1)(2,1,0)(1,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(4,2)=B(0,0,0)(1,1,1)(2,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(4,2)(5,0)(5,2)=B(0,0,0)(1,1,1)(2,1,1)(3,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,0)=B(0,0,0)(1,1,1)(2,2,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(5,0)=B(0,0,0)(1,1,1)(2,2,0)(2,2,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(5,2)=B(0,0,0)(1,1,1)(2,2,0)(3,1,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(6,0)=B(0,0,0)(1,1,1)(2,2,0)(3,2,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,1)=B(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,0,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,0)(5,1)(6,1)=B(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,0,0)(5,0,0)\)

\((0,0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,3)=B(0,0,0)(1,1,1)(2,2,0)(3,2,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,2)=B(0,0,0)(1,1,1)(2,2,1)\)

\((0,0)(1,1)(2,2)(3,2)(4,3)=B(0,0,0)(1,1,1)(2,2,1)(3,0,0)\)