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

0, 5, 6, 75, 8, 83, 857142, 875, 8, 9, 90, 916, 923076, 9285714, 93, 9375, 9411764705882352, 94, 947368421052631578, 95, 952380, 954, 9565217391304347826086, 9583, 96, 9615384, 962, 96428571, 9655172413793103448275862068, 96, 967741935483870, 96875, 96, 97058823529411764, 9714285, 972, 972

Offset: 0

Author

Doug Bell, Jun 23 2015

Keywords

• nonn
• base

Comments

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).

Other than n = (36,37) the only repeated terms appear to follow one of the following two patterns for the larger value of n:

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).

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).

Have verified that there are no other repeating terms up to n = 10^6.

Examples

			a(1)=5 (1/2=0.5), a(2)=6 (2/3=0.6666...=6), a(3)=75 (3/4=0.75=75).
		

Links

• Doug Bell, Table of n, a(n) for n = 0..1000

Crossrefs

Subsequences A156703, A235589.

Cf. A036275, A060284.

Programs

Mathematica

Array[FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] &, 37] (* Michael De Vlieger, Aug 18 2015 *)