Jake Foth

Structural Theory of Recamán's Sequence and Its Subtraction Indices

Recamán's sequence (OEIS A005132) is generated by subtracting the step number when the resulting nonnegative value is new, and adding otherwise. This preprint develops an exact structural theory of its addition/subtraction branch word and of its successful-subtraction indices (OE...

Structural Theory of Recamán's Sequence and Its Subtraction Indices

Recamán's sequence (OEIS A005132) is generated by subtracting the step number when the resulting nonnegative value is new, and adding otherwise. This preprint develops an exact structural theory of its addition/subtraction branch word and of its successful-subtraction indices (OE...

A Fibonacci-Automatic Set Whose Enumerator Image Is Not Context-Free

Let S be the Golden Exclusion set, let T be its complement, let B(n)=∣T∩[1,n]∣, and let q(k) be the increasing enumeration of T. This paper studies R2=q(T), the first nontrivial self-enumerator image in the associated Kimberling dispersion. On the regular landmark family Z(pa)=(1...

Perfect Squares in Fibonacci-Product Bands: An Exact Length Spectrum and Minimum-Root Sequence

Let F0=0,F1=1. For even m≥6 and L≥4, define I1(m,L)=[FmFm+L,FmFm+L+FL−2−1] and I2(m,L)=[FmFm+L+FL−1,FmFm+L+FL+1−1]. These intervals arise from Golden Exclusion choice words with a maximal two-copy stable period-4 tail C(⌊φf2⌋)=p(1100)r, r≥2. We prove that a square f2, with f not...

From Residue Dynamics to Arithmetic Fibers: The Pronic Structure of A392243 and Its Elliptic Sister Fiber

This paper develops the exact arithmetic structure behind a modulo-threshold pseudo-Recamán recurrence and the signed square-block sequence OEIS A392243. At the critical threshold, the canonical orbit is proved to be the absolute value of the signed sum, with an exact folded pron...