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A309698 Digits of the 4-adic integer 3^(1/3).

%I A309698 #20 Aug 14 2019 08:32:20
%S A309698 3,2,3,1,1,0,3,3,1,0,2,0,3,3,0,3,1,3,0,1,1,3,0,3,3,3,3,3,1,0,3,2,0,2,
%T A309698 0,0,1,2,3,2,0,3,1,0,1,1,1,2,1,2,0,1,0,1,3,2,2,1,1,1,3,2,2,0,3,3,3,0,
%U A309698 3,0,0,0,3,0,2,3,3,0,3,2,1,2,1,2,2,1,0,0,0,2,0,1,3,0
%N A309698 Digits of the 4-adic integer 3^(1/3).
%H A309698 Seiichi Manyama, <a href="/A309698/b309698.txt">Table of n, a(n) for n = 0..10000</a>
%H A309698 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A309698 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + b(n-1)^3 - 3 mod 4^n for n > 1, then a(n) = (b(n+1) - b(n))/4^n.
%o A309698 (PARI) N=100; Vecrev(digits(lift((3+O(2^(2*N)))^(1/3)), 4), N)
%o A309698 (Ruby)
%o A309698 def A309698(n)
%o A309698   ary = [3]
%o A309698   a = 3
%o A309698   n.times{|i|
%o A309698     b = (a + a ** 3 - 3) % (4 ** (i + 2))
%o A309698     ary << (b - a) / (4 ** (i + 1))
%o A309698     a = b
%o A309698   }
%o A309698   ary
%o A309698 end
%o A309698 p A309698(100)
%Y A309698 Digits of the k-adic integer (k-1)^(1/(k-1)): this sequence (k=4), A309699 (k=6), A309700 (k=8), A225458 (k=10).
%Y A309698 Cf. A225404, A290563, A322931.
%K A309698 nonn,base
%O A309698 0,1
%A A309698 _Seiichi Manyama_, Aug 13 2019