Problem 44628. The other half of the Fibonacci sequence

Created by David Verrelli

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://mathworld.wolfram.com/FibonacciNumber.html\"><w:r><w:t>\"Fibonacci sequence\"</w:t></w:r></w:hyperlink><w:r><w:t> — F = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...] — appeared in Indian mathematical expositions from</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>circa</w:t></w:r><w:r><w:t> 700 CE or earlier, and in the writings of Leonardo of Pisa (a.k.a. \"Fibonacci\")</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>circa</w:t></w:r><w:r><w:t> 1202 CE.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>This sequence can be defined by</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>F(n+2) = F(n+1) + F(n)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>in which F(1) = 1, F(2) = 1, F(3) = 2, ....</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Later in history, it was recognised that F(0) = 0. Of course, this still satisfies the formula in bold above [for n=0]: F(2) = F(1) + F(0).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Your job in this Cody Problem is to 'create history'(?) by extending this sequence to</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>negative values of n</w:t></w:r><w:r><w:t>, to discover the missing half of this sequence!</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>EXAMPLE:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>If n=-1, then F(-1) must be 1, to ensure that F(1) = F(0) + F(-1) — thus satisfying the formula in bold above.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>You are only required to provide outputs for n &lt; 3 that can be represented by an</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/int64.html\"><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>int64</w:t></w:r></w:hyperlink><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>. To enforce this, your output needs to be of this data type.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

The <a href = "http://mathworld.wolfram.com/FibonacciNumber.html">"Fibonacci sequence"</a> — 
F = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...] — appeared in Indian mathematical expositions from circa 700 CE or earlier, and in the writings of Leonardo of Pisa (a.k.a. "Fibonacci") circa 1202 CE.This sequence can be defined byF(n+2) = F(n+1) + F(n)in which F(1) = 1, F(2) = 1, F(3) = 2, ....Later in history, it was recognised that F(0) = 0. Of course, this still satisfies the formula in bold above [for n=0]: F(2) = F(1) + F(0).Your job in this Cody Problem is to 'create history'(?) by extending this sequence to negative values of n, to discover the missing half of this sequence!EXAMPLE:If n=-1, then F(-1) must be 1, to ensure that F(1) = F(0) + F(-1) — thus satisfying the formula in bold above.You are only required to provide outputs for n &lt; 3 that can be represented by an <a href = "https://au.mathworks.com/help/matlab/ref/int64.html"><tt>int64</tt></a> <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>. To enforce this, your output needs to be of this data type.

The <http://mathworld.wolfram.com/FibonacciNumber.html "Fibonacci sequence"> — 
F = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...] — appeared in Indian mathematical expositions from _circa_ 700 CE or earlier, and in the writings of Leonardo of Pisa (a.k.a. "Fibonacci") _circa_ 1202 CE.

This sequence can be defined by

*F(n+2) = F(n+1) + F(n)*

in which F(1) = 1, F(2) = 1, F(3) = 2, ....

Later in history, it was recognised that F(0) = 0.  Of course, this still satisfies the formula in bold above [for n=0]:  F(2) = F(1) + F(0).

Your job in this Cody Problem is to 'create history'(?) by extending this sequence to _negative values of n_, to discover the missing half of this sequence!

EXAMPLE:

If n=-1, then F(-1) must be 1, to ensure that F(1) = F(0) + F(-1) — thus satisfying the formula in bold above.

You are only required to provide outputs for n < 3 that can be represented by an <https://au.mathworks.com/help/matlab/ref/int64.html |int64|> <https://au.mathworks.com/help/matlab/numeric-types.html data type>. To enforce this, your output needs to be of this data type.

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Last Solution submitted on Nov 20, 2021

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Solution 1584952

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David Verrelli on 24 Jun 2019

Good job on commenting your code!

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