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

%I A262114 #29 Feb 16 2025 08:33:27
%S A262114 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,
%T A262114 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,
%U A262114 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
%N A262114 Irregular triangle read by rows: row b (b >= 2) gives periodic part of digits of the base-b expansion of 1/5.
%C A262114 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).
%C A262114 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.
%C A262114 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}.
%C A262114 b == 0 (mod 5): 1 (terminating)
%C A262114 b == 1 (mod 5): 1 (purely recurrent)
%C A262114 b == 2 (mod 5): 4 (purely recurrent)
%C A262114 b == 3 (mod 5): 4 (purely recurrent)
%C A262114 b == 4 (mod 5): 2 (purely recurrent)
%C A262114 The expansion of 1/5 has a full-length period 4 when base b is a primitive root of p = 5.
%C A262114 Digits of 1/5 for the following bases:
%C A262114 2    0, 0, 1, 1
%C A262114 3    0, 1, 2, 1
%C A262114 4    0, 3
%C A262114 5*   1
%C A262114 6    1
%C A262114 7    1, 2, 5, 4
%C A262114 8    1, 4, 6, 3
%C A262114 9    1, 7
%C A262114 10*  2
%C A262114 11   2
%C A262114 12   2, 4, 9, 7
%C A262114 13   2, 7, 10, 5
%C A262114 14   2, 11
%C A262114 15*  3
%C A262114 16   3
%C A262114 17   3, 6, 13, 10
%C A262114 18   3, 10, 14, 7
%C A262114 19   3, 15
%C A262114 20*  4
%C A262114 ...
%C A262114 Asterisks above denote terminating expansion; all other entries are digits of purely recurrent reptends.
%C A262114 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.
%C A262114 Entries for b == 0 (mod 5) (i.e., integer multiples of 5) appear at 11, 23, 35, ..., every 12th term thereafter.
%D A262114 U. Dudley, Elementary Number Theory, 2nd ed., Dover, 2008, pp. 119-126.
%D A262114 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 6th ed., Oxford Univ. Press, 2008, pp. 138-148.
%D A262114 Oystein Ore, Number Theory and Its History, Dover, 1988, pp. 311-325.
%H A262114 Michael De Vlieger, <a href="/A262114/b262114.txt">Table of n, a(n) for n = 2..10000</a>
%H A262114 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DecimalPeriod.html">Decimal Period</a>.
%H A262114 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RepeatingDecimal.html">Repeating Decimal</a>.
%F A262114 Conjectures from _Colin Barker_, Oct 09 2015: (Start)
%F A262114 a(n) = 2*a(n-12) - a(n-24) for n>24.
%F A262114 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).
%F A262114 (End)
%e A262114 For b = 8, 1/5 = .14631463..., thus 1, 4, 6, 3 are terms in the sequence.
%e A262114 For b = 10, 1/5 = .2, thus 2 is a term in the sequence.
%e A262114 For b = 13, 1/5 = .27a527a5..., thus 2, 7, 10, 5 are terms in the sequence.
%t A262114 RotateLeft[Most@ #, Last@ #] &@ Flatten@ RealDigits[1/5, #] & /@ Range[2, 38] // Flatten (* _Michael De Vlieger_, Sep 11 2015 *)
%Y A262114 Cf. A004526 Digits of expansions of 1/2.
%Y A262114 Cf. A026741 Full Reptends of 1/3.
%Y A262114 Cf. A130845 Digits of expansions of 1/3 (eliding first 2 terms).
%Y A262114 Cf. A262115 Digits of expansions of 1/7.
%K A262114 nonn,base,tabf
%O A262114 2,7
%A A262114 _Michael De Vlieger_, Sep 11 2015