A021023 Decimal expansion of 1/19.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

The 18-digit cycle 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2 in this sequence and the others based on nineteenths gives the successive digits of the smallest integer which is doubled, quadrupled and octupled when the last three digits in turn are moved from the right hand end to the left hand end. For example, 842105263157894736 is eight times 105263157894736842. - Ian Duff, Jan 07 2009, Jan 12 2009

The magic square that uses the decimals of 1/19 is fully magic. 383 has the same property (see A021387). For other such primes see A072359. - Michel Marcus, Sep 02 2015

Since 19 is prime and the cycle of its reciprocal's base 10 digits is 19 - 1 long, 19 is a full reptend prime (A001913). - Alonso del Arte, Mar 21 2020

References

• Martin Gardner, Cyclic numbers, Mathematical Circus, Chapter 10, p. 172, of the 1992 Mathematical Association of America edition.
• GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.

Links

• Wikipedia, Prime reciprocal magic square
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,-1,1).

Crossrefs

Cf. A021387, A072359.

Programs

Mathematica

Prepend[First@ RealDigits[N[1/19, 120]], 0] (* Michael De Vlieger, Sep 02 2015 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,-1,1},{0,5,2,6,3,1,5,7,8,9},100] (* or *) PadRight[{},100,{0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1}] (* Harvey P. Dale, Jan 23 2021 *)

PARI

default(realprecision,2000);1/19.0 \\ Anders Hellström, Sep 02 2015

Scala

def longDivRecip(n: Int, places: Int = 100): List[Int] = {
  val pow10 = Math.pow(10, Math.ceil(Math.log10(Math.abs(n)))).toInt
  val digits = new scala.collection.mutable.ListBuffer[Int]()
  var quotient = pow10; var remainder = 0
  while (digits.size < places) {
    remainder = quotient % n; quotient /= n; digits += quotient
    quotient = remainder * 10
  }
  digits.toList
}
0 :: longDivRecip(19) // Alonso del Arte, Mar 20 2020

Formula

G.f.: -x*(x^8 + x^7 + 2*x^6 + 4*x^5 - 2*x^4 - 3*x^3 + 4*x^2 - 3*x + 5)/((x - 1)*(x + 1)*(x^2 - x + 1)*(x^6 - x^3 + 1)). - Colin Barker, Aug 15 2012