cp's OEIS Frontend

A259299 The decimal expansion of n/(n+1) until it terminates or repeats, shown without the decimal point.

%I A259299 #18 Aug 18 2015 11:39:11
%S A259299 0,5,6,75,8,83,857142,875,8,9,90,916,923076,9285714,93,9375,
%T A259299 9411764705882352,94,947368421052631578,95,952380,954,
%U A259299 9565217391304347826086,9583,96,9615384,962,96428571,9655172413793103448275862068,96,967741935483870,96875,96,97058823529411764,9714285,972,972
%N A259299 The decimal expansion of n/(n+1) until it terminates or repeats, shown without the decimal point.
%C A259299 The first occurrence of a repeated term where a(n) = a(n+1) is for a(35) and a(36), both of which equal 972.  This results from two different repeating decimals with different length repeating periods but the same non-repeating plus repeating digits, namely 35/36 = .972222... = 972 (repeating period of 1) and 36/37 = .972972... = 972 (repeating period of 3).
%C A259299 Other than n = (36,37) the only repeated terms appear to follow one of the following two patterns for the larger value of n:
%C A259299   First pattern: for n >= 111, where all digits of n are 1: 111, 1111, 11111, ... and a(n-1) = a(n) = 990, 9990, 99990, ... with the repeating decimal for ((n-1)/n, n/(n+1)) of (.9909090..., .990990990...), (.9990990990..., .999099909990...), (.9999099909990..., .999909999099990...).  Where d is the number of digits in a(n), the repeating period for the decimal values is (d-1, d).
%C A259299   Second pattern: for n >= 10101, where the digits of n alternate between 0 and 1, with a final digit of 1: 10101, 1010101, 101010101, ... and a(n-1) = a(n) = 999900, 9999900, 99999900, ... with the repeating decimal for ((n-1)/n, n/(n+1)) of (.99990099009900..., .999900999900999900...), (.99999009990099900..., .999990099999009999900...), (.99999900999900999900..., .999999009999990099999900...).   Where d is the number of digits in a(n), the repeating period for the decimal values is (d-2, d).
%C A259299 Have verified that there are no other repeating terms up to n = 10^6.
%H A259299 Doug Bell, <a href="/A259299/b259299.txt">Table of n, a(n) for n = 0..1000</a>
%e A259299 a(1)=5 (1/2=0.5), a(2)=6 (2/3=0.6666...=6), a(3)=75 (3/4=0.75=75).
%t A259299 Array[FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] &, 37] (* _Michael De Vlieger_, Aug 18 2015 *)
%Y A259299 Subsequences A156703, A235589.
%Y A259299 Cf. A036275, A060284.
%K A259299 nonn,base
%O A259299 0,2
%A A259299 _Doug Bell_, Jun 23 2015