Alex Ashburn

Iterated string replacements from 00: limits, discrepancy bounds and lengths in conjectures of Kimberling

In 2017 Kimberling entered in the OEIS the words obtained by iterating string replacements such as 00->0101, 1->010 from the word 00, for 63 such maps, with more than 200 conjectures on the positions a(n) of the letters in the limiting words and on the lengths of the iterates. A...

Discrepancy of the words ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ and twenty-two conjectures of Kimberling

For an irrational r > 0 and an integer k ≥ 1, the word s_k(n) = ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ takes only the values 0 and 1, each with density 1/2. In 2019 Kimberling added to the OEIS eleven such words, for r ∈ {φ, √2, √3, e} and k ≤ 4, together with the 22 sequences a(n) listing...

Exact discrepancy of letter positions in fixed points of binary morphisms, with an application to conjectures of Kimberling

In 2017 Kimberling added to the On-Line Encyclopedia of Integer Sequences a large family of conjectures of the form L < nr − a(n) < U for all n ≥ 1. Here a(n) is the position of the n-th 0 (or 1) in the fixed point, or limiting word, of a morphism on {0,1}, and 1/r is the frequen...

Exact discrepancy of letter positions in fixed points of binary morphisms, with an application to conjectures of Kimberling

In 2017 Kimberling added to the On-Line Encyclopedia of Integer Sequences a large family of conjectures of the form L < nr − a(n) < U for all n ≥ 1. Here a(n) is the position of the n-th 0 (or 1) in the fixed point, or limiting word, of a morphism on {0,1}, and 1/r is the frequen...

Kimberling's linear complementary equations: automatic structure and exact bounds

In 2018 Kimberling added to the On-Line Encyclopedia of Integer Sequences the solutions of complementary equations such as a(n) = 2b(n−1) − b(n−2) + 2n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and b is the increasing sequence of positive integers not in a. For 21 of these s...

Kimberling's linear complementary equations: automatic structure and exact bounds

In 2018 Kimberling added to the On-Line Encyclopedia of Integer Sequences the solutions of complementary equations such as a(n) = 2b(n−1) − b(n−2) + 2n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and b is the increasing sequence of positive integers not in a. For 21 of these s...

On three conjectures of Kimberling concerning the array ⌊kφ^n⌋

Let φ be the golden ratio and let R_n = {⌊kφ^n⌋ : k ≥ 1} be the n-th row of the array T(n,k) = ⌊kφ^n⌋ (OEIS A128440). In 2022 Kimberling conjectured that the rows R_{2n−1} and R_{2n} are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they...

Discrepancy of the words ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ and twenty-two conjectures of Kimberling

For an irrational r > 0 and an integer k ≥ 1, the word s_k(n) = ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ takes only the values 0 and 1, each with density 1/2. In 2019 Kimberling added to the OEIS eleven such words, for r ∈ {φ, √2, √3, e} and k ≤ 4, together with the 22 sequences a(n) listing...

Exact discrepancy of letter positions in fixed points of binary morphisms, with an application to conjectures of Kimberling

In 2017 Kimberling added to the On-Line Encyclopedia of Integer Sequences a large family of conjectures of the form L < nr − a(n) < U for all n ≥ 1. Here a(n) is the position of the n-th 0 (or 1) in the fixed point, or limiting word, of a morphism on {0,1}, and 1/r is the frequen...