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A021113
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Decimal expansion of 1/109.
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5
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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, 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, 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
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OFFSET
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0,3
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COMMENTS
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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.
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.
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)
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LINKS
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FORMULA
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Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/10^(k+1). - Amiram Eldar, Feb 05 2022
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EXAMPLE
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0.00917431192660550458715596330275229357798165137614...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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