A021113 Decimal expansion of 1/109.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

From Paul Curtz, Feb 23 2012: (Start)

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)

Examples

			0.00917431192660550458715596330275229357798165137614...
		

Links

• "Xochipilli", Le Webinet des curiosités:les meilleures recettes de Kaprekar (in French)
• Guillaume Yoda, Dictionnaire des nombres - 109 (in French)
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A000045, A018481, A021093, A064737.

Programs

Mathematica

RealDigits[1/109, 10, 100][[1]] (* Alonso del Arte, Feb 11 2012 *)

PARI

1/109. \\ Charles R Greathouse IV, Feb 13 2012

Formula

Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/10^(k+1). - Amiram Eldar, Feb 05 2022