76 lines
3.1 KiB
Plaintext
76 lines
3.1 KiB
Plaintext
<!-- The following <math> tag doesn't render properly as it does correctly on Wikipedia.
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It generates some bold red error messages.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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:<math>
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L(n) =
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\begin{cases}
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1 & \mbox{if } n = 0 \\
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1 & \mbox{if } n = 1 \\
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L(n - 1) + L(n - 2) + 1 & \mbox{if } n > 1 \\
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\end{cases}
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</math>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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(end of comment) -->
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''Leonardo numbers'' are also known as the ''Leonardo series''.
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The '''Leonardo numbers''' are a sequence of numbers defined by:
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<big> L(0) = 1 [1<sup>st</sup> equation] </big>
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<big> L(1) = 1 [2<sup>nd</sup> equation] </big>
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<big> L(n) = L(n-1) + L(n-2) + 1 [3<sup>rd</sup> equation] </big>
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─── also ───
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<big> L(n) = 2 * Fib(n+1) - 1 [4<sup>th</sup> equation] </big>
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:::: where the '''+ 1''' will herein be known as the ''add'' number.
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:::: where the '''FIB''' is the [[wp:Fibonacci number|Fibonacci number]]s.
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This task will be using the 3<sup>rd</sup> equation (above) to calculate the Leonardo numbers.
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[[wp:Edsger W. Dijkstra|Edsger W. Dijkstra]] used Leonardo numbers as an integral part of
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his [[wp:smoothsort|smoothsort]] [[wp:algorithm|algorithm]].
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<!-- The following <math> tag doesn't render properly as it does correctly on Wikipedia:.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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:<math>1,\;1,\;3,\;5,\;9,\;15,\;25,\;41,\;67,\;109,\;177,\;287,\;465,\;753,\;1219,\;1973,\;3193,\;5167,\;8361, \ldots</math>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-->
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The first few Leonardo numbers are:
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''' 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 ··· '''
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;Task:
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::* show the 1<sup>st</sup> '''25''' Leonardo numbers, starting at '''L(0)'''.
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::* allow the first two Leonardo numbers to be specified [for '''L(0)''' and '''L(1)'''].
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::* allow the ''add'' number to be specified ('''1''' is the default).
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::* show the 1<sup>st</sup> '''25''' Leonardo numbers, specifying '''0''' and '''1''' for '''L(0)''' and '''L(1)''', and '''0''' for the ''add'' number.
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(The last task requirement will produce the Fibonacci numbers.)
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Show all output here on this page.
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;Related tasks:
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* [[Fibonacci number]]
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* [[Fibonacci n-step number sequences ]]
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;See also:
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* [[wp:Leonardo number|Wikipedia, Leonardo numbers]]
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* [[wp:Fibonacci number|Wikipedia, Fibonacci numbers]]
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* [[oeis:A001595|OEIS Leonardo numbers]]
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<br><br>
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