World!OfNumbers |
WON plate 228 | |
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[ A wealth of information is given in OEIS A000045. The Fibonacci number is described in https://www.geeksforgeeks.org/find-index-given-fibonacci-number-constant-time/ as
\(Fn = {1 \over sqrt(5)} * (a^n - b^n) \) where
\(a = {1 + sqrt(5) \over 2} \) and \(\color{red}b = {1 - sqrt(5) \over 2} \)
On neglecting \(b^n\) which is very small because of the large value of n, we keep
\(Fn = {1 \over \sqrt{5}} * ({1 + \sqrt{5} \over 2})^n \)
Let us express this equation in function of index n using logarithm acrobacy
\(log(Fn) = log({1 \over \sqrt{5}}) + n * log({1 + \sqrt{5} \over 2}) \)
\(log(Fn) = -0.8047189562... +\ n\ *\ 0.4812118250... \)
\(log(Fn)\ +\ 0.8047189562... =\ n\ *\ 0.4812118250... \)
\(n\ =\ {{log(Fn)\ +\ 0.8047189562...}\over{0.4812118250...}} \)
\(n\ =\ {2.0780869212...\ *\ log(Fn)\ +\ 1.6722759381...} \)
The palindromes only show up when the accuracy is reduced to 6 decimals
\(n = round ( 2.0\color{blue}{78087}\color{black}{\ *\ log(Fn)\ +\ 1.}\color{blue}{672276} \color{black}{)} \)
where round means round to the nearest integer. What more is there to tell about these two peculiar palindromes ?
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A000228 Prime Curios! Prime Puzzle Wikipedia 228 Le nombre 228 |

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