Leonardo number

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The Leonardo numbers are a sequence of numbers given by the recurrence:

 
  L(n) = 
  \begin{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[1] used them as an integral part of his smoothsort algorithm,[2] and also analyzed them in some detail.[3]

Computing a second-order recurrence relation recursively and without memoization requires L(n) computations for the n-th item of the series.

Relation to Fibonacci numbers[edit]

The Leonardo numbers are related to the Fibonacci numbers by the relation L(n) = 2 F(n+1) - 1, n \ge 0.

From this relation it is straightforward to derive a closed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers:

L(n) = 2 \frac{\varphi^{n+1} - \psi^{n+1}}{\varphi - \psi}- 1 = \frac{2}{\sqrt 5} \left(\varphi^{n+1} - \psi^{n+1}\right) - 1 = 2F(n+1) - 1

where the golden ratio \varphi = \left(1 + \sqrt 5\right)/2 and \psi = \left(1 - \sqrt 5\right)/2 are the roots of the quadratic polynomial x^2 - x - 1 = 0.

The first few Leonardo numbers are

1,\;1,\;3,\;5,\;9,\;15,\;25,\;41,\;67,\;109,\;177,\;287,\;465,\;753,\;1219,\;1973,\;3193,\;5167,\;8361, \ldots (sequence A001595 in OEIS)

References[edit]

  1. ^ EWD797
  2. ^ Dijkstra, Edsger W. Smoothsort – an alternative to sorting in situ (EWD-796a). E.W. Dijkstra Archive. Center for American History, University of Texas at Austin.  (original; transcription)
  3. ^ EWD796a

External links[edit]