What Is the N‑bonacci Sequence? Exploring Generalized Fibonacci Numbers

Fibonacci Sequence

The classic Fibonacci sequence starts with 0 and 1, and each subsequent term is the sum of the two preceding terms, producing an infinite series with many remarkable properties, including a ratio sequence that converges to the golden ratio.

N‑bonacci Sequence

The N‑bonacci sequence generalizes the Fibonacci rule by summing the previous N terms to obtain the next term. It starts with N‑1 zeros and a single 1 . For example, a 3‑bonacci sequence begins with three zeros and a 1, and each term is the sum of the preceding three terms.

Similarly, a 4‑bonacci sequence starts with four zeros and a 1, and each term is the sum of the previous four terms.

Characteristics of N‑bonacci

Initial condition : each N‑bonacci sequence begins with N‑1 zeros followed by a 1.

Recurrence relation : every term equals the sum of the preceding N terms.

Infinite sequence : like the Fibonacci sequence, the N‑bonacci sequence continues indefinitely.

N‑bonacci Constant

Just as the Fibonacci ratio converges to the golden ratio, the ratio of consecutive terms of an N‑bonacci sequence converges to a specific constant called the N‑bonacci constant. For N=2 (the classic Fibonacci), the constant is ≈1.61803; for N=3 it is ≈1.83928; for N=4 it is ≈1.92756; for N=5 it is ≈1.96595.

Table of the first five N‑bonacci constants:

N  →  N‑bonacci constant 1  →  1 2  →  1.61803 3  →  1.83928 4  →  1.92756 5  →  1.96595

Infinite N‑bonacci Sequence

Considering an “infinite‑bonacci” where the number of preceding terms tends to infinity, the sequence starts with infinitely many zeros followed by a 1. The resulting terms are powers of 2 (1, 1, 2, 4, 8, 16, …) and the ratio of consecutive terms is constantly 2, making the infinite‑bonacci constant equal to 2.

Observing the progression from the Fibonacci constant (≈1.618) to the 3‑bonacci constant (≈1.83928) and finally to the infinite‑bonacci constant (2) shows a monotonic increase, prompting interest in concise methods for computing these constants.

Readers are invited to leave comments with thoughts or questions.