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A225406
Digits of the 10-adic integer 9^(1/3).
8
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
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
KEYWORD
nonn,base
AUTHOR
Aswini Vaidyanathan, May 07 2013
STATUS
approved