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Search: id:A097348
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| A097348 |
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First n terms give number of digits of Fibonacci[10^n] (except that it can be off by 1). |
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+0
2
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| 2, 0, 8, 9, 8, 7, 6, 4, 0, 2, 4, 9, 9, 7, 8, 7, 3, 3, 7, 6, 9, 2, 7, 2, 0, 8, 9, 2, 3, 7, 5, 5, 5, 4, 1, 6, 8, 2, 2, 4, 5, 9, 2, 3, 9, 9, 1, 8, 2, 1, 0, 9, 5, 3, 5, 3, 9, 2, 8, 7, 5, 6, 1, 3, 9, 7, 4, 1, 0, 4, 8, 5, 3, 4, 9, 6, 7, 4, 5, 9, 6, 3, 2, 7, 7, 6, 5, 8, 5, 5, 6, 2, 3, 5, 1, 0, 3, 5, 3, 5, 1, 4, 5, 0
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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This is an example of a highly compressed sequence. As a result, it can be off by one. The uncompressed version goes like this: 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, 208987640, 2089876403, ... (see A068070).. Fibonacci[10] = 55 has 2 digits, Fibonacci[100] = 354224848179261915075 has 21 digits and so on.
Equal to decimal expansion of ArcCsch[2]/log[10] = 0.2089876..., Eric Weisstein (eric(AT)weisstein.com), Aug 24, 2004 [Corrected by Hugh Warrington (hughwarringtonNOSPAM(AT)gmail.com), Mar 03 2005]
Considering the very good approximation F(n)=5^(-1/2)*phi^n, the number of digits of F(10^n) is given by int(log10(F(10^n)))=int(-1/2*log10(5)+10^n*log10(phi)). Similary L(n) tends to phi^n, so the number of digits of L(10^n) is given by int(10^n*log10(phi)). Both numbers can differ at most by 1. F(n) and L(n) denote the Fibonacci and Lucas numbers, resp. - Christoph Pacher (christoph.pacher(AT)arcs.ac.at), Nov 22 2006
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LINKS
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Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number
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EXAMPLE
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Fibonacci[10^9] has 208987640 decimal digits
Fibonacci[10^21] has 208987640249978733769 decimal digits
Fibonacci[10^27] has 208987640249978733769272089 decimal digits
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MATHEMATICA
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FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]
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CROSSREFS
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Cf. A000045, A068070.
Sequence in context: A019658 A021483 A011015 this_sequence A106193 A085814 A009524
Adjacent sequences: A097345 A097346 A097347 this_sequence A097349 A097350 A097351
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KEYWORD
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easy,nonn,cons
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AUTHOR
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Ed Pegg Jr (ed(AT)mathpuzzle.com), Aug 06 2004
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