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 Vm={v0<⋯<vm−1}, let gj,n(m) count outputs equal to vj, aggregated over all full bracketings on n variables and all mn valuations. The root-pair refinement produces Ni,j(m)(n)=∑r=1n−1gi,r(m)gj,n−r(m), with generating functions Ni,j(m)(x)=Gi(m)(x)Gj(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 binomm+12 distinct root-pair generating functions. We catalogue the factories for m=4,5,…,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).
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BibTeX
@misc{yildiz2026sequence,
title = {{Sequence Factory: Families of Integer Sequences Generated by Gödel Implication}},
author = {Volkan Yildiz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20028},
eprint = {2610.20028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20028},
}
RIS
TY - UNPB
TI - Sequence Factory: Families of Integer Sequences Generated by Gödel Implication
AU - Volkan Yildiz
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20028
UR - https://arxiv.org/abs/2610.20028
AB - 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<⋯<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)=∑_r=1^n−1g_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 binomm+12 distinct root-pair generating functions. We catalogue the factories for m=4,5,…,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).
ER -