A309698 - OEIS

%I #20 Aug 14 2019 08:32:20

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

%H Seiichi Manyama, <a href="/A309698/b309698.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) + b(n-1)^3 - 3 mod 4^n for n > 1, then a(n) = (b(n+1) - b(n))/4^n.

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

%o (Ruby)

%o def A309698(n)

%o   ary = [3]

%o   a = 3

%o   n.times{|i|

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

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

%o     a = b

%o   }

%o   ary

%o end

%o p A309698(100)

%Y 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 Cf. A225404, A290563, A322931.

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Aug 13 2019