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A105509 Numbers n such that 9 is the leading digit of the n-th Fibonacci number in decimal representation. 9
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A008963(a(n)) = 9; A105519(a(n)) = A105519(a(n) - 1) + 1.

Comment from Jonathan Vos Post, 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 0's are appended to the numerator and a 9 to the beginning and end of the denominator...."

LINKS

Table of n, a(n) for n=1..51.

M. Bicknell-Johnson, A generalized magic trick from Fibonacci: Designer decimals, College Mathematics Journal 35(March):125-126, 2004.

O-Y. Chan and J. Smoak, More designer decimals: The integers and their geometric extensions College Mathematics Journal 37(November):355-363, 2006.

Ivars Peterson, Designer Decimals, Science News, Week of Nov 04 2006; Vol. 170, No. 19.

J. Smoak, and T.J. Osler, A magic trick from Fibonacci. College Mathematics Journal, 34 (2003):58-60.

FORMULA

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, Dec 23 2006

a(n) ~ kn by the equidistribution theorem, where k = 1/(1 - log(9)/log(10)) = 21.8543.... - Charles R Greathouse IV, Oct 07 2016

EXAMPLE

a(10)=21: A008963(212) = A000030(A000045(212)) =

A000030(90343046356137747723758225621187571439538669) = 9.

MATHEMATICA

Select[Range@ 1200, First@ IntegerDigits@ Fibonacci@ # == 9 &] (* Michael De Vlieger, Aug 21 2016 *)

PROG

(PARI) is(n)=digits(fibonacci(n))[1]==9 \\ Charles R Greathouse IV, Oct 07 2016

CROSSREFS

Cf. A000030, A000045, A072711, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105508.

Sequence in context: A190741 A294073 A086119 * A337024 A219316 A317818

Adjacent sequences:  A105506 A105507 A105508 * A105510 A105511 A105512

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

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Last modified May 10 05:10 EDT 2021. Contains 343748 sequences. (Running on oeis4.)