Leonardo number

Leonardo number

The Leonardo numbers are a sequence of numbers given by:: L(n):= egin{cases} 1 & mbox{if } n = 0; \ 1 & mbox{if } n = 1; \ L(n-1)+L(n-2)+1 & mbox{if } n > 1. \ end{cases}

Edsger W. Dijkstra [ [http://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html EWD797] ] used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail. [ [http://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD796a.html EWD796a] ]

They are related to the Fibonacci numbers by the relation L(n) = 2 imes F(n+1) - 1 (following the convention that F(0)=0).

The first few Leonardo numbers are: 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, … OEIS|id=A001595

References


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