Sequence Factory: Families of Integer Sequences Generated by Gödel Implication

Abstract

A finite Gödel chain generates not one integer sequence but a large family of them. The combinatorial mechanism is developed in a companion paper; here the sequences themselves are the primary objects. For $V_m=\{v_0<\cdots<v_{m-1}\}$, let $g_{j,n}^{(m)}$ count outputs equal to $v_j$, aggregated over all full bracketings on $n$ variables and all $m^n$ valuations. The root-pair refinement produces $N_{i,j}^{(m)}(n)=\sum_{r=1}^{n-1}g_{i,r}^{(m)}g_{j,n-r}^{(m)}$, with generating functions $N_{i,j}^{(m)}(x)=G_i^{(m)}(x)G_j^{(m)}(x)$. We prove that the initial coefficients of these products uniquely identify the unordered root-value pair. Consequently, for each fixed $m$, there are exactly $\binom{m+1}{2}$ distinct root-pair generating functions. We catalogue the factories for $m=4,5,\ldots,10$, giving the first ten terms of every base sequence and every symmetric root-pair sequence. This range contains 49 base sequences and 210 symmetric pair classes, yielding 259 indexed sequence families before additional aggregates. The catalogue provides reproducible data for integer-sequence analysis and comparison with the On-Line Encyclopedia of Integer Sequences (OEIS).