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Search: id:A105509
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| A105509 |
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Numbers n such that 9 is the leading digit of the n-th Fibonacci number in decimal representation. |
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+0
10
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| 16, 35, 59, 83, 102, 126, 150, 169, 193, 212, 236, 260, 279, 303, 327, 346, 370, 394, 413, 437, 461, 480, 504, 528, 547, 571, 595, 614, 638, 657, 681, 705, 724, 748, 772, 791, 815, 839, 858, 882, 906, 925, 949, 973, 992, 1016, 1040, 1059, 1083, 1102, 1107
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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A008963(a(n)) = 9; A105519(a(n)) = A105519(a(n) - 1) + 1.
Comment from Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 23 2006: Peterson says: "Calculate 100/89 = 1.1235955056... This fraction generates the first five Fibonacci numbers before blurring into other digits. ... 10000/9899 = 1.0102030508132134559046368... generates the first 10 Fibonacci numbers (using two digits per number). 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number). ... in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator...."
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REFERENCES
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Bicknell-Johnson, M. 2004. A generalized magic trick from Fibonacci: Designer decimals. College Mathematics Journal 35(March):125-126.
Chan, O-Y. and J. Smoak. 2006. More designer decimals: The integers and their geometric extensions. College Mathematics Journal 37(November):355-363.
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LINKS
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Smoak, J. and T.J. Osler, A magic trick from Fibonacci. College Mathematics Journal, 34 (2003):58-60.
Ivars Peterson, Designer Decimals, Science News, Week of Nov. 4, 2006; Vol. 170, No. 19.
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FORMULA
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n such that d(n+5)-d(n) = 2 for d(n) = floor(1 + log base 10(F(n))) and F(n) = n-th Fibonacci number = A000045(n). - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 23 2006
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EXAMPLE
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a(10)=21: A008963(212) = A000030(A000045(212)) =
A000030(90343046356137747723758225621187571439538669) = 9.
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CROSSREFS
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Cf. A000030, A000045, A072711, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105508.
Sequence in context: A104910 A161441 A086119 this_sequence A070588 A109287 A066112
Adjacent sequences: A105506 A105507 A105508 this_sequence A105510 A105511 A105512
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KEYWORD
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nonn,base
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 11 2005
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