巨大数研究 Wiki
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巨大数研究 Wiki

弱ハイパーペア数列[1]は、公太朗[2]が2020年8月23日に公開した表記である。

定義

\begin{aligned}弱ハイパーペア数列数 &=\text{WHPair}^{\circ 86}(86)\\ \text{WHPair}(n)&=\text{expand}((0,0)(1,\omega)(1,1)[n])\\ \text{expand}([n]) &=n\\ \text{expand}(\textbf{S}[n]) &=\begin{cases}\text{expand}((S_{00},S_{01})(S_{10},S_{11})\cdots(S_{(X-1)0},S_{(X-1)1})[10^n]) &(\text{if}\ S_{X0}=0)\\ \text{expand}((S_{00},S_{01})(S_{10},S_{11})\cdots(S_{X0},n)[10^n]) &(\text{if}\ S_{X1}=\omega)\\ \text{expand}(\textbf{G}\textbf{B}^0\textbf{B}^1\cdots\textbf{B}^n[10^n])&(\text{otherwise}) \end{cases}\\ \textbf{S}&=(S_{00},S_{01})(S_{10},S_{11})\cdots(S_{X0},S_{X1})\\ \textbf{G}&=(S_{00},S_{01})(S_{10},S_{11})\cdots(S_{(r-1)0},S_{(r-1)1})\\ \textbf{B}^m &=\textbf{B}^m_r\textbf{B}^m_{r+1}\cdots\textbf{B}^m_{X-1}\\ \textbf{B}^m_x&=(S_{x0}+m\Delta_0,S_{x1}+m\Delta_{x1})\\ \Delta_0 &=\begin{cases}0&(\text{if}~S_{X1}=0)\\ S_{X0}- S_{r0}&(\text{otherwise}) \end{cases}\\ \Delta_{x1}&=\begin{cases}0&(\text{if}~\nexists a.r=p^{\circ a}_1(x)\lor S_{X1}=0)\\ S_{X1}- S_{r1}- 1&(\text{otherwise}) \end{cases}\\ r &=\begin{cases}p_0(X) &(\text{if}~S_{X1}=0)\\ p_1(X) &(\text{if}~\text{diff}(0)=1)\\ p^{\circ\gamma}_1(X)&(\text{otherwise}) \end{cases}\\ \gamma &=\begin{cases}\min\{k \mid 0=S_{p^{\circ k}_1(X)1}\}&(\text{if}~\nexists a.\text{diff}(a) \lt \text{diff}(0))\\ \min\{k \mid \text{diff}(k) \lt \text{diff}(0) \}&(\text{otherwise}) \end{cases}\\ \text{diff}(x) &=S_{p^{\circ x}_1(X)1}-S_{p^{\circ x+1}_1(X)1}\\ p_0(x) &=\max\{k \mid S_{k0}\lt S_{x0}\land k \lt x \}\\ p_1(x) &=\max\{k \mid S_{k1}\lt S_{x1}\land \exists a. k=p^{\circ a}_0(x) \}\\ \end{aligned}

出典

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