A105509 Numbers m such that 9 is the leading digit of the m-th Fibonacci number in decimal representation.

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

Offset: 1

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

• nonn
• base

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...."

Examples

			a(10)=21: A008963(212) = A000030(A000045(212)) =
A000030(90343046356137747723758225621187571439538669) = 9.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

m such that d(m+5)-d(m) = 2 for d(m) = floor(1 + log_10(F(m))) and F(m) = m-th Fibonacci number = A000045(m). - Jonathan Vos Post, Dec 23 2006

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