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

%I A225406 #42 Aug 13 2019 08:35:22
%S A225406 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,
%T A225406 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,
%U A225406 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
%N A225406 Digits of the 10-adic integer 9^(1/3).
%H A225406 Robert Israel, <a href="/A225406/b225406.txt">Table of n, a(n) for n = 0..10000</a>
%F A225406 p = ...660569, q = A225409 = ...339431, p + q = 0. - _Seiichi Manyama_, Aug 04 2019
%F A225406 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
%e A225406        9^3 == 9 (mod 10).
%e A225406       69^3 == 9 (mod 10^2).
%e A225406      569^3 == 9 (mod 10^3).
%e A225406      569^3 == 9 (mod 10^4).
%e A225406    60569^3 == 9 (mod 10^5).
%e A225406   660569^3 == 9 (mod 10^6).
%p A225406 op([1,3],padic:-rootp((x)^3  -9,  10, 101)); # _Robert Israel_, Aug 04 2019
%o A225406 (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)))
%o A225406 (PARI) Vecrev(digits(truncate(-(-9+O(10^100))^(1/3)))) \\ _Seiichi Manyama_, Aug 04 2019
%o A225406 (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
%o A225406 (Ruby)
%o A225406 def A225406(n)
%o A225406   ary = [9]
%o A225406   a = 9
%o A225406   n.times{|i|
%o A225406     b = (a + 3 * (a ** 3 - 9)) % (10 ** (i + 2))
%o A225406     ary << (b - a) / (10 ** (i + 1))
%o A225406     a = b
%o A225406   }
%o A225406   ary
%o A225406 end
%o A225406 p A225406(100) # _Seiichi Manyama_, Aug 13 2019
%Y A225406 Cf. A309600.
%Y A225406 Digits of 10-adic integers:
%Y A225406 A153042 ((-1/9)^(1/3));
%Y A225406 A225409 (  (-9)^(1/3));
%Y A225406 A225412 ( (1/9)^(1/3)).
%K A225406 nonn,base
%O A225406 0,1
%A A225406 _Aswini Vaidyanathan_, May 07 2013