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A309722 Digits of the 4-adic integer (1/3)^(1/3).

%I A309722 #19 Aug 17 2019 05:06:38
%S A309722 3,0,3,2,1,1,0,1,2,2,2,0,1,1,0,0,1,0,0,0,1,2,3,2,2,3,2,3,3,1,1,2,0,1,
%T A309722 3,0,0,2,3,2,2,2,0,0,0,0,0,3,2,0,2,0,2,0,0,2,3,2,3,2,2,3,3,2,2,2,0,2,
%U A309722 3,1,0,0,3,3,2,3,3,3,0,3,1,3,2,3,2,2,1,2,0,3,2,0,2,3,0,0,2,0,3,3,0
%N A309722 Digits of the 4-adic integer (1/3)^(1/3).
%H A309722 Seiichi Manyama, <a href="/A309722/b309722.txt">Table of n, a(n) for n = 0..10000</a>
%H A309722 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A309722 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + 3 * (3 * b(n-1)^3 - 1) mod 4^n for n > 1, then a(n) = (b(n+1) - b(n))/4^n.
%o A309722 (PARI) N=100; Vecrev(digits(lift((1/3+O(2^(2*N)))^(1/3)), 4), N)
%o A309722 (Ruby)
%o A309722 def A309722(n)
%o A309722   ary = [3]
%o A309722   a = 3
%o A309722   n.times{|i|
%o A309722     b = (a + 3 * (3 * a ** 3 - 1)) % (4 ** (i + 2))
%o A309722     ary << (b - a) / (4 ** (i + 1))
%o A309722     a = b
%o A309722   }
%o A309722   ary
%o A309722 end
%o A309722 p A309722(100)
%Y A309722 Digits of the k-adic integer (1/(k-1))^(1/(k-1)): this sequence (k=4), A309723 (k=6), A309724 (k=8), A225464 (k=10).
%Y A309722 Cf. A225411, A309698.
%K A309722 nonn,base
%O A309722 0,1
%A A309722 _Seiichi Manyama_, Aug 14 2019