Sequence of Fibonacci Numbers ending in Index

Theorem

Let $F_k$ denote the $k$th Fibonacci number.

For all $k \in \Z$, let $F_k$ be expressed in decimal notation.

The sequence of integers $\sequence n$ such that $F_n$ ends in $n$ starts:

$0, 1, 5, 25, 29, 41, 49, 61, 65, 85, 89, 101, 125, 145, 149, 245, 265, 365, 385, 485, 505, 601, \ldots$

This sequence is A000350 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Sources

• 1966: Gerard R. Deily: Terminal Digit Coincidences Between Fibonacci Numbers and Their Indices (The Fibonacci Quarterly Vol. 4: pp. 151 – 157)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$