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A309613 Digits of the 10-adic integer (13/9)^(1/3).

%I A309613 #22 Aug 12 2019 10:17:05
%S A309613 3,9,6,9,6,6,2,5,6,6,5,7,4,8,2,6,1,5,2,3,5,4,9,5,1,2,1,5,3,6,9,7,1,6,
%T A309613 7,5,5,3,7,6,4,9,5,3,1,7,9,8,0,4,4,5,7,1,2,1,3,3,4,9,1,0,1,7,6,1,9,8,
%U A309613 0,4,1,3,7,6,7,1,2,2,0,1,1,5,4,9,2,5,2,8,9,7,5,0,9,1,4,5,4,9,7,3
%N A309613 Digits of the 10-adic integer (13/9)^(1/3).
%H A309613 Seiichi Manyama, <a href="/A309613/b309613.txt">Table of n, a(n) for n = 0..10000</a>
%F A309613 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + 3 * (9 * b(n-1)^3 - 13) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.
%e A309613        3^3 == 7      (mod 10).
%e A309613       93^3 == 57     (mod 10^2).
%e A309613      693^3 == 557    (mod 10^3).
%e A309613     9693^3 == 5557   (mod 10^4).
%e A309613    69693^3 == 55557  (mod 10^5).
%e A309613   669693^3 == 555557 (mod 10^6).
%o A309613 (PARI) N=100; Vecrev(digits(lift(chinese(Mod((13/9+O(2^N))^(1/3), 2^N), Mod((13/9+O(5^N))^(1/3), 5^N)))), N)
%o A309613 (Ruby)
%o A309613 def A309613(n)
%o A309613   ary = [3]
%o A309613   a = 3
%o A309613   n.times{|i|
%o A309613     b = (a + 3 * (9 * a ** 3 - 13)) % (10 ** (i + 2))
%o A309613     ary << (b - a) / (10 ** (i + 1))
%o A309613     a = b
%o A309613   }
%o A309613   ary
%o A309613 end
%o A309613 p A309613(100)
%Y A309613 Cf. A178769, A309600, A309608.
%K A309613 nonn,base
%O A309613 0,1
%A A309613 _Seiichi Manyama_, Aug 10 2019