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A259299
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The decimal expansion of n/(n+1) until it terminates or repeats, shown without the decimal point.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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a(1)=5 (1/2=0.5), a(2)=6 (2/3=0.6666...=6), a(3)=75 (3/4=0.75=75).
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MATHEMATICA
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Array[FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] &, 37] (* Michael De Vlieger, Aug 18 2015 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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