Pentagonal Numbers in the Fibonacci Sequence

Ming Luo

A chapter in Applications of Fibonacci Numbers, 1996, pp 349-354 from Springer

Abstract: Abstract It is well known that the numbers of the form $$ \frac{1}{2}m\left( {3m - 1} \right), $$ m is a positive integer, are called pentagonal numbers. In this paper we consider the question of how many pentagonal numbers there are in the Fibonacci sequence $$ {F_n}{ + _2} = {F_n} + {F_n},\,{F_0} = 0,\,{F_1} = 1 $$ where n ranges over all integers. We find there are exactly two such numbers, which are $$ {F_{ \pm 1}} = {F_2} = 1 $$ and $$ {F_{ \pm 5}} = 5. $$

Date: 1996
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DOI: 10.1007/978-94-009-0223-7_29

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