This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A021113 #55 Feb 12 2025 11:24:04
%S A021113 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 A021113 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 A021113 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 A021113 Decimal expansion of 1/109.
%C A021113 From _Paul Curtz_, Feb 23 2012: (Start)
%C A021113 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 A021113 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 A021113 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 A021113 "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 A021113 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)
%H A021113 <a href="/index/Rec#order_55">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).
%F A021113 Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/10^(k+1). - _Amiram Eldar_, Feb 05 2022
%e A021113 0.00917431192660550458715596330275229357798165137614...
%t A021113 RealDigits[1/109, 10, 100][[1]] (* _Alonso del Arte_, Feb 11 2012 *)
%o A021113 (PARI) 1/109. \\ _Charles R Greathouse IV_, Feb 13 2012
%Y A021113 Cf. A000045, A018481, A021093, A064737.
%K A021113 nonn,cons,easy
%O A021113 0,3
%A A021113 _N. J. A. Sloane_