# Bashicu Matrix Calculator

Initial variable:

Maximum length: n increment:

Detailed output:

Bashicu matrix calculator shows calculation process of Bashicu matrix. Set initial variable in the form of BM[n], where BM is a sequence expression of Bashicu matrix, n is a natural number, and press "Go" button. See some examples below. You can also download and use offline version of this program with milder limit on input variables.

(Japanese) バシク行列計算機は、バシク行列の計算過程を表示します。Initial variable に、初期値 BM[n] (BM はバシク行列の数列表記) を入力して Go ボタンを押して下さい。Maximum length は計算を終了する数列の長さ（行列の列数）、n increment は n の値をどのように変化させるかを指定します。下に例を示します。オフラインバージョンをインストールすることもできます。

## Bashicu matrix

Bashicu matrix system is a notation designed to produce large numbers. Bashicu matrix is a matrix such as $% $ where all elements are nonnegative integers. The matrix can be written in the form of $(a_{11},a_{21})(a_{12},a_{22})(a_{13},a_{23})$; sequence of transpose of each column. With an algorithm invented by Bashicu in 2014 and updated as shown below in version history, Bashich matrix BM works as a function from a natural number n to a natural number BM[n] (provided that the calculation ends), and written as (0,0)(1,1)(1,1)[3]. It is known that 2-row matrix, pair sequence, can be approximated with Hardy function. When the function is approximated with Hardy function, the matrix itself represents the ordinal of the Hardy function, and therefore can be written as:

Official definition of the algorithm of Bashicu matrix is in the source code of C program of this site, Bashicu matrix calculator. Human readable definition and analysis of Bashicu matrix system is available at the entry at googology wiki.

## Versions of BM

Version Status Author Year
Version 1 (BM1)NT Bashicu 2014
Version 2 (BM2)NT Bashicu 2016
Version 2.1 (BM2.1)NT koteitan 2018
Version 2.2 (BM2.2)B koteitan 2018
Version 2.3 (BM2.3)B koteitan 2018
Version 3 (BM3)NT Bashicu 2018
Version 3.1 (BM3.1)B Nish 2018
Version 3.2 (BM3.2)B Nish 2018
Version 3.3 (BM3.3)B rpakr, Ecl1psed 2019
Version 4 (BM4)B Bashicu 2018

Status column shows if (0,0,0,...,0)(1,1,1,...,1)[n] always terminates.

• T: It is proved that it always terminates. There is no such version yet.
• B: Believed to terminate but there is no proof.
• NT: It is proved that it does not always terminate.

See original definitions of each version for detail.