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

%I #19 Aug 17 2019 05:06:38

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

%H Seiichi Manyama, <a href="/A309722/b309722.txt">Table of n, a(n) for n = 0..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.

%F 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 (PARI) N=100; Vecrev(digits(lift((1/3+O(2^(2*N)))^(1/3)), 4), N)

%o (Ruby)

%o def A309722(n)

%o ary = [3]

%o a = 3

%o n.times{|i|

%o b = (a + 3 * (3 * a ** 3 - 1)) % (4 ** (i + 2))

%o ary << (b - a) / (4 ** (i + 1))

%o a = b

%o }

%o ary

%o end

%o p A309722(100)

%Y 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 Cf. A225411, A309698.

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Aug 14 2019