Jake Foth

Between Consecutive Squares: The Arithmetic of A392243

Between Consecutive Squares develops the arithmetic of OEIS sequence A392243, defined by the signed sum a(n) = sum_{i=1}^n i(-1)^{ceil(sqrt(i))}, whose signs change immediately after each perfect square. The book begins from an elementary closed formula for the sequence and follo...

Between Consecutive Squares: The Arithmetic of A392243

Between Consecutive Squares develops the arithmetic of OEIS sequence A392243, defined by the signed sum a(n) = sum_{i=1}^n i(-1)^{ceil(sqrt(i))}, whose signs change immediately after each perfect square. The book begins from an elementary closed formula for the sequence and follo...

Fibonacci Sixth Powers in a Signed Square-Block Sum

At first glance, A392243 comes from a simple rule: add the positive integers in order, changing the sign whenever the running index crosses into a new square-number block. This creates a sequence with a surprisingly rich internal structure. The question at the center of this pape...

Extrema of a(n)/n^(log_phi 2) for the Fibbinary Numbers on Intervals Between Consecutive Fibonacci Numbers

This note studies the Fibbinary numbers, OEIS A003714, through the map that transfers the digits of the Zeckendorf representation of an integer to the corresponding binary place values. Writing κ=log⁡φ2\kappa=\log_\varphi 2, it determines the exact maximum and minimum of a(n)/nκa...

The Odd-Even Digit Walk: Tie-Broken Symmetry, Zero-Area Returns, and Decimal Renewal Geometry

We study the Odd-Even Digit Walk, the integer recurrence obtained by taking steps of magnitude nn and choosing the sign from the difference between the sums of the even-valued and odd-valued decimal digits of the current state, with ties assigned to addition. Although the rule di...

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...