Two properties of the generalized sequence \(\{W_n\}\) relevant to recurring decimal. (Q2715985)
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scientific article; zbMATH DE number 1600954
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Two properties of the generalized sequence \(\{W_n\}\) relevant to recurring decimal. |
scientific article; zbMATH DE number 1600954 |
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20 July 2005
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Fibonacci number
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Lucas number
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decimal expansion
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rational generating function
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0.84947544
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Two properties of the generalized sequence \(\{W_n\}\) relevant to recurring decimal. (English)
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The author considers sequences of integers \(\{W_n\}\) given by \(W_0=a\), \(W_1=b\), and \(W_{n+2}=pW_{n+1}+qW_n\) (\(a,b,p,q\) are integers). This is a generalization of Fibonacci numbers \(F_n\) (\(p=q=1\), \(a=0\), \(b=1\)) and Lucas numbers. Using the rational OGF (ordinary generating function) of \(\{W_n\}\), the author proves (Corollary 2) the triviality that \(\sum _{i\geq 0}F_i/10^{i+1}=1/89\). Another eight corollaries and two theorems on this level are given.
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