A262114 Irregular triangle read by rows: row b (b >= 2) gives periodic part of digits of the base-b expansion of 1/5.

0, 0, 1, 1, 0, 1, 2, 1, 0, 3, 1, 1, 1, 2, 5, 4, 1, 4, 6, 3, 1, 7, 2, 2, 2, 4, 9, 7, 2, 7, 10, 5, 2, 11, 3, 3, 3, 6, 13, 10, 3, 10, 14, 7, 3, 15, 4, 4, 4, 8, 17, 13, 4, 13, 18, 9, 4, 19, 5, 5, 5, 10, 21, 16, 5, 16, 22, 11, 5, 23, 6, 6, 6, 12, 25, 19, 6, 19, 26, 13, 6, 27, 7, 7, 7, 14, 29, 22, 7, 22

Offset

2,7

Comments

The number of terms associated with a particular value of b are cyclical: 4, 4, 2, 1, 1, repeat. This is because the values are associated with b (mod 5), starting with 2 (mod 5).

The expansion of 1/5 either terminates after one digit when b == 0 (mod 5) or is purely recurrent in all other cases of b (mod 5), since 5 is prime and must either divide or be coprime to b.

The period for purely recurrent expansions of 1/5 must be a divisor of Euler's totient of 5 = 4, i.e., one of {1, 2, 4}.

b == 0 (mod 5): 1 (terminating)

b == 1 (mod 5): 1 (purely recurrent)

b == 2 (mod 5): 4 (purely recurrent)

b == 3 (mod 5): 4 (purely recurrent)

b == 4 (mod 5): 2 (purely recurrent)

The expansion of 1/5 has a full-length period 4 when base b is a primitive root of p = 5.

Digits of 1/5 for the following bases:

2 0, 0, 1, 1

3 0, 1, 2, 1

4 0, 3

5* 1

6 1

7 1, 2, 5, 4

8 1, 4, 6, 3

9 1, 7

10* 2

11 2

12 2, 4, 9, 7

13 2, 7, 10, 5

14 2, 11

15* 3

16 3

17 3, 6, 13, 10

18 3, 10, 14, 7

19 3, 15

20* 4

...

Asterisks above denote terminating expansion; all other entries are digits of purely recurrent reptends.

Each entry associated with base b with more than one term has a second term greater than the first except for b = 2, where the first two terms are 0, 0.

Entries for b == 0 (mod 5) (i.e., integer multiples of 5) appear at 11, 23, 35, ..., every 12th term thereafter.

References

U. Dudley, Elementary Number Theory, 2nd ed., Dover, 2008, pp. 119-126.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 6th ed., Oxford Univ. Press, 2008, pp. 138-148.

Oystein Ore, Number Theory and Its History, Dover, 1988, pp. 311-325.

Links

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Eric Weisstein's World of Mathematics, Decimal Period.

Eric Weisstein's World of Mathematics, Repeating Decimal.

Formula

Conjectures from Colin Barker, Oct 09 2015: (Start)

a(n) = 2*a(n-12) - a(n-24) for n>24.

G.f.: x^3*(x^19 +x^18 +x^17 +2*x^16 +2*x^15 +x^14 +2*x^13 +3*x^12 +2*x^11 +x^10 +x^9 +x^8 +3*x^7 +x^5 +2*x^4 +x^3 +x +1) / (x^24 -2*x^12 +1).

(End)

Example

For b = 8, 1/5 = .14631463..., thus 1, 4, 6, 3 are terms in the sequence.
For b = 10, 1/5 = .2, thus 2 is a term in the sequence.
For b = 13, 1/5 = .27a527a5..., thus 2, 7, 10, 5 are terms in the sequence.

Mathematica

RotateLeft[Most@ #, Last@ #] &@ Flatten@ RealDigits[1/5, #] & /@ Range[2, 38] // Flatten (* Michael De Vlieger, Sep 11 2015 *)

Crossrefs

Cf. A004526 Digits of expansions of 1/2.

Cf. A026741 Full Reptends of 1/3.

Cf. A130845 Digits of expansions of 1/3 (eliding first 2 terms).

Cf. A262115 Digits of expansions of 1/7.

Sequence in context: A167192 A335435 A363048 * A320780 A029312 A287352

Adjacent sequences: A262111 A262112 A262113 * A262115 A262116 A262117

Keyword

nonn,base,tabf

Author

Michael De Vlieger, Sep 11 2015

Status

approved