%I #49 Feb 05 2022 06:41:22
%S 0,0,9,1,7,4,3,1,1,9,2,6,6,0,5,5,0,4,5,8,7,1,5,5,9,6,3,3,0,2,7,5,2,2,
%T 9,3,5,7,7,9,8,1,6,5,1,3,7,6,1,4,6,7,8,8,9,9,0,8,2,5,6,8,8,0,7,3,3,9,
%U 4,4,9,5,4,1,2,8,4,4,0,3,6,6,9,7,2,4,7,7,0,6,4,2,2,0,1,8,3,4,8,6,2,3,8,5,3,2,1,1
%N Decimal expansion of 1/109.
%C From _Paul Curtz_, Feb 23 2012: (Start)
%C The sequence of digits is periodic with period length 108. A feature of the period reading from the least significant digit back to the most significant digit is (see the blogspot link and A064737) that it "contains" the single-digit of every Fibonacci subsequence if the digits are added with carry of the previous sum. A064737 starts with the A000045 sequence, and then 5+8 = (1)3, 3+8+1=(1)2. "Every" Fibonacci sequence means (as illustrated in the blog) that one could also start from seeds like 6 and 7, or 7 and 8.
%C Similar observations are made for the digits of 1/89 in A021093, but following a Fibonacci pattern while reading in the other direction, starting with the most significant digits.
%C The frequency distribution of the digits 0 to 9 among the 108 digits (which sum to 486) of the period is well-balanced: 10, 11, 11, 11, 11, 11, 11, 11, 11, 10. If one sums over each 2nd, each 3rd, each 6th, each 9th or each 18th digit of the period, one gets 1/2, 1/3, 1/6, 1/9 and 1/18 of 486; again a feature of balance in the digits. There is a half-period in the sense that a(n) + a(n+54) = 9. (End)
%H "Xochipilli", <a href="http://webinet.blogspot.com/2009/01/kaprekarez-vous.html">Le Webinet des curiosités:les meilleures recettes de Kaprekar</a> (in French)
%H Guillaume Yoda, <a href="https://web.archive.org/web/20190319101620/http://yoda.guillaume.pagesperso-orange.fr/N100a500/Nb109.htm">Dictionnaire des nombres - 109</a> (in French)
%F Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/10^(k+1). - _Amiram Eldar_, Feb 05 2022
%e 0.00917431192660550458715596330275229357798165137614...
%t RealDigits[1/109, 10, 100][[1]] (* _Alonso del Arte_, Feb 11 2012 *)
%o (PARI) 1/109. \\ _Charles R Greathouse IV_, Feb 13 2012
%Y Cf. A000045, A018481, A021093, A064737.
%K nonn,cons,easy
%O 0,3
%A _N. J. A. Sloane_