A021093 Decimal expansion of 1/89.

0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn
• cons
• easy

Comments

Note the strange resemblance to the Fibonacci numbers (A000045). In fact 1/89 = Sum_{j>=0} Fibonacci(j)/10^(j+1). (In the same way, the Lucas numbers sum up to 120/89.) - Johan Claes, Jun 11 2004

In the Red Zen reference, the decimal expansion of 1/89 and its relation to the Fibonacci sequence is discussed; also primes of the form floor((1/89)*10^n) are given for n = 3, 5 and 631. - Jason Earls, May 28 2007

The 44-digit cycle 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 4, 4, 7, 1, 9 in this sequence, and the others based on eighty-ninths, give the successive digits of the smallest integer that is multiplied by nine when the final digit is moved from the right hand end to the left hand end. - Ian Duff, Jan 09 2009

Generalization (since Fibonacci(j+2) = Fibonacci(j+1) + Fibonacci(j)):

1/89 = Sum_{j>=0} Fibonacci(j) / 10^(j+1), (this sequence)

1/9899 = Sum_{j>=0} Fibonacci(j) / 100^(j+1),

1/998999 = Sum_{j>=0} Fibonacci(j) / 1000^(j+1),

1/99989999 = Sum_{j>=0} Fibonacci(j) / 10000^(j+1),

...

1 / ((10^k)^2 - (10^k)^1 - (10^k)^0) = 1 / (10^(2k) - 10^k - 1) =

Sum_{j>=0} Fibonacci(j) / (10^k)^(j+1), k >= 1.

- Daniel Forgues, Oct 28 2011, May 04 2013

Generalization (since 11^(j+1) = 11 * 11^j):

1/89 = Sum_{j>=0} 11^j / 100^(j+1), (this sequence)

1/989 = Sum_{j>=0} 11^j / 1000^(j+1),

1/9989 = Sum_{j>=0} 11^j / 10000^(j+1),

1/99989 = Sum_{j>=0} 11^j / 100000^(j+1),

...

1 / ((10^k)^1 - 11 (10^k)^0) = 1 / (10^k - 11) =

Sum_{j>=0}^ 11^j / (10^k)^(j+1), k >= 2.

- Daniel Forgues, Oct 28 2011, May 04 2013

More generally, Sum_{k>=0} F(k)/x^k = x/(x^2 - x - 1) (= g.f. of signed Fibonacci numbers -A039834, because of negative powers). This yields 10/89 for x=10. Dividing both sides by x=10 gives the constant A021093, cf. first comment. - M. F. Hasler, May 07 2014

Replacing x with a power of 10 (positive or negative exponent) in an o.g.f. gives similar constants for many sequences. For example, setting x=1/1000 in (1 - sqrt(1 - 4*x)) / (2*x) gives 1.001002005014042132... (cf. A000108). - Joerg Arndt, May 11 2014

References

• Jason Earls, Red Zen, Lulu Press, NY, 2007, pp. 47-48. ISBN: 978-1-4303-2017-3.
• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 66.

Links

• Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 10.
• 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,-1,1).

Crossrefs

Cf. A000045, A001020.

Programs

Maple

Digits:=100; evalf(1/89); # Wesley Ivan Hurt, May 08 2014

Mathematica

RealDigits[1/89, 10, 100, -1] (* Wesley Ivan Hurt, May 08 2014 *)

PARI

1/89. \\ Charles R Greathouse IV, Dec 05 2011