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

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, 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, 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

Offset

0,1

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Wikipedia, Hensel's Lemma.

Formula

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.

Prog

(PARI) N=100; Vecrev(digits(lift((1/3+O(2^(2*N)))^(1/3)), 4), N)
(Ruby)
def A309722(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + 3 * (3 * a ** 3 - 1)) % (4 ** (i + 2))
    ary << (b - a) / (4 ** (i + 1))
    a = b
  }
  ary
end
p A309722(100)

Crossrefs

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).

Cf. A225411, A309698.

Sequence in context: A171911 A180193 A229964 * A070298 A024938 A332715

Adjacent sequences: A309719 A309720 A309721 * A309723 A309724 A309725

Keyword

nonn,base

Author

Seiichi Manyama, Aug 14 2019

Status

approved