A225406 Digits of the 10-adic integer 9^(1/3).

9, 6, 5, 0, 6, 6, 3, 4, 8, 6, 6, 0, 4, 8, 5, 4, 5, 9, 4, 5, 1, 1, 9, 4, 0, 6, 0, 8, 1, 3, 7, 0, 6, 6, 9, 4, 8, 3, 9, 9, 3, 0, 2, 4, 2, 0, 3, 5, 9, 8, 6, 5, 5, 0, 9, 6, 7, 7, 4, 8, 0, 7, 4, 6, 1, 0, 3, 2, 9, 8, 5, 8, 2, 1, 5, 7, 0, 9, 0, 9, 8, 8, 1, 6, 0, 6, 8, 6, 0, 3, 9, 5, 0, 9, 9, 5, 6, 5, 3, 7

Offset

0,1

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

p = ...660569, q = A225409 = ...339431, p + q = 0. - Seiichi Manyama, Aug 04 2019

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 9, b(n) = b(n-1) + 3 * (b(n-1)^3 - 9) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n. - Seiichi Manyama, Aug 13 2019

Example

       9^3 == 9 (mod 10).
      69^3 == 9 (mod 10^2).
     569^3 == 9 (mod 10^3).
     569^3 == 9 (mod 10^4).
   60569^3 == 9 (mod 10^5).
  660569^3 == 9 (mod 10^6).

Maple

op([1, 3], padic:-rootp((x)^3  -9,  10, 101)); # Robert Israel, Aug 04 2019

Prog

(PARI) n=0; for(i=1, 100, m=9; for(x=0, 9, if(((n+(x*10^(i-1)))^3)%(10^i)==m, n=n+(x*10^(i-1)); print1(x", "); break)))
(PARI) Vecrev(digits(truncate(-(-9+O(10^100))^(1/3)))) \\ Seiichi Manyama, Aug 04 2019
(PARI) N=100; Vecrev(digits(lift(chinese(Mod((9+O(2^N))^(1/3), 2^N), Mod((9+O(5^N))^(1/3), 5^N)))), N) \\ Seiichi Manyama, Aug 04 2019
(Ruby)
def A225406(n)
  ary = [9]
  a = 9
  n.times{|i|
    b = (a + 3 * (a ** 3 - 9)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A225406(100) # Seiichi Manyama, Aug 13 2019

Crossrefs

Cf. A309600.

Digits of 10-adic integers:

A153042 ((-1/9)^(1/3));

A225409 ( (-9)^(1/3));

A225412 ( (1/9)^(1/3)).

Sequence in context: A154183 A200138 A196772 * A347330 A343947 A367709

Adjacent sequences: A225403 A225404 A225405 * A225407 A225408 A225409

Keyword

nonn,base

Author

Aswini Vaidyanathan, May 07 2013

Status

approved