Abstract

In this note, we study Fibonacci-like sequences that are defined by the recurrence S_{k} = a, S_{k+1} = b, S_{n+2} ≡ S_{n+1} +S_{n} (mod n+2) for all n ≥ k, where k, a, b ∈ ℕ, 0 ≤ a < k, 0 ≤ b < k + 1, and (a, b) ≠ (0, 0). We will show that the number α = 0.S_{k}S_{k+1}S_{k+2} ⋯ is irrational. We also propose a conjecture on the pattern of the sequence {S_{n}}_{n≥k}.