A262115 -id:A262115 - cp's OEIS frontend

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

Michael De Vlieger, Sep 11 2015

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.

			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.

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.

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.

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.

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

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)

cp's OEIS Frontend

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

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