Extended Fibonacci numbers

The well known Fibonacci numbers are defined recursively as follows: $F_0 = 0$ , $F_1 = 1$ , $F_i = F_{i-1} + F_{i-2}$ for $i \ge 2$ . The first Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, ….

Let us generalize the Fibonacci numbers. For every pair of natural numbers $a$ and $b$ , define the sequence $S(a, b) = [f_0, f_1, \dots]$ as $f_0 = a$ , $f_1 = b$ , $f_i = f_{i-1} + f_{i-2}$ for $i \ge 2$ . Note that $S(0, 1)$ is the traditional Fibonacci sequence.

You are given a natural number $n$ . Please compute how many pairs $(a, b)$ exist such that $S(a, b)$ has a $i \ge 3$ where $f_i = n$ . For instance, for $n = 2$ there are exactly three such sequences: $S(0, 1) = [0, 1, 1, 2, \dots]$ , $S(1, 0) = [1, 0, 1, 1, 2, \dots]$ , and $S(2, 0) = [2, 0, 2, 2, \dots]$ .

Input

Input consists of several cases, each with a different natural number $n$ between 1 and $10^6$ .

Output

For every $n$ , print the number of pairs $(a, b)$ such that $n$ appears at a position $i \ge 3$ in $S(a, b)$ .

Hint

Depending on your solution, Cassini’s identity could be useful: $F_{i-1} \cdot F_{i+1} - F_i^2 = (-1)^i$ .

Problem information

Author: Salvador Roura

Generation: 2026-01-25T10:26:51.803Z