| abstract | - Pair sequence number is a the output of a program made by BashicuHyudora and posted at a googology-related thread of the Japanese site BBS 2ch.net in 2014. The algorithm of the program is called the pair sequence system, a weak version of Bashicu's matrix system by the same creator. It is supposed to calculate number comparable to \(f_{\vartheta(\Omega_\omega)+1}(10)\). It is an extension of a system named the primitive sequence system, still by the same author, which generates a number approaching \(f_{\varepsilon_0+1}(10)\). A pair sequence is a finite sequence of pairs of nonnegative integers, for example (0,0)(1,1)(2,2)(3,3)(3,2). A pair sequence P works as a function from natural numbers to natural numbers, (though we write P[n] rather than P(n)), for example \(n \mapsto (0,0)(1,1)(2,2)(3,3)(3,2)[n]\) is a function. The function P[n] is usually approximated with a function of the form \(H_\alpha\) from the Hardy hierarchy (we note \(P = \alpha\)). For example, \((0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2))\).
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