Smallest Positive Integer with 5 Fibonacci Partitions
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Theorem
The smallest positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways is $24$.
Proof
\(\displaystyle 1\) | \(=\) | \(\displaystyle 1\) | $1$ way | ||||||||||
\(\displaystyle 2\) | \(=\) | \(\displaystyle 2\) | $1$ way | ||||||||||
\(\displaystyle 3\) | \(=\) | \(\displaystyle 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 2 + 1\) | $2$ ways | ||||||||||
\(\displaystyle 4\) | \(=\) | \(\displaystyle 3 + 1\) | $1$ way | ||||||||||
\(\displaystyle 5\) | \(=\) | \(\displaystyle 5\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 3 + 2\) | $2$ ways | ||||||||||
\(\displaystyle 6\) | \(=\) | \(\displaystyle 5 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 3 + 2 + 1\) | $2$ ways | ||||||||||
\(\displaystyle 7\) | \(=\) | \(\displaystyle 5 + 2\) | $1$ way | ||||||||||
\(\displaystyle 8\) | \(=\) | \(\displaystyle 8\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 5 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 5 + 2 + 1\) | $3$ ways | ||||||||||
\(\displaystyle 9\) | \(=\) | \(\displaystyle 8 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 5 + 3 + 1\) | $2$ ways | ||||||||||
\(\displaystyle 10\) | \(=\) | \(\displaystyle 8 + 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 5 + 3 + 2\) | $2$ ways | ||||||||||
\(\displaystyle 11\) | \(=\) | \(\displaystyle 8 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 2 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 5 + 3 + 2 + 1\) | $3$ ways | ||||||||||
\(\displaystyle 12\) | \(=\) | \(\displaystyle 8 + 3 + 1\) | $1$ way | ||||||||||
\(\displaystyle 13\) | \(=\) | \(\displaystyle 13\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 3 + 2\) | $3$ ways | ||||||||||
\(\displaystyle 14\) | \(=\) | \(\displaystyle 13 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 3 + 2 + 1\) | $3$ ways | ||||||||||
\(\displaystyle 15\) | \(=\) | \(\displaystyle 13 + 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 2\) | $2$ ways | ||||||||||
\(\displaystyle 16\) | \(=\) | \(\displaystyle 13 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 2 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 2 + 1\) | $4$ ways | ||||||||||
\(\displaystyle 17\) | \(=\) | \(\displaystyle 13 + 3 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 3 + 1\) | $2$ ways | ||||||||||
\(\displaystyle 18\) | \(=\) | \(\displaystyle 13 + 5\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 3 + 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 3 + 2\) | $3$ ways | ||||||||||
\(\displaystyle 19\) | \(=\) | \(\displaystyle 13 + 5 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 3 + 2 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 8 + 5 + 3 + 2 + 1\) | $3$ ways | ||||||||||
\(\displaystyle 20\) | \(=\) | \(\displaystyle 13 + 5 + 2\) | $1$ way | ||||||||||
\(\displaystyle 21\) | \(=\) | \(\displaystyle 21\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 8\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 5 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 5 + 2 + 1\) | $4$ ways | ||||||||||
\(\displaystyle 22\) | \(=\) | \(\displaystyle 21 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 8 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 5 + 3 + 1\) | $3$ ways | ||||||||||
\(\displaystyle 23\) | \(=\) | \(\displaystyle 21 + 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 8 + 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 5 + 3 + 2\) | $3$ ways | ||||||||||
\(\displaystyle 24\) | \(=\) | \(\displaystyle 21 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 21 + 2 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 8 + 3\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 8 + 2 + 1\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 13 + 5 + 3 + 2 + 1\) | $5$ ways |
This sequence is A000119 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
$\blacksquare$
Sources
- 1966: David A. Klarner: Representations of $n$ as a Sum of Distinct Elements from Special Sequences (The Fibonacci Quarterly Vol. 4: 289 – 306)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$