A225412 - OEIS

%I #54 Aug 13 2019 08:36:44

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

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

%U 6,1,4,0,2,0,9,6,3,6,6,1,9,1,9,9,7,4,9,8,8,7,7,3,0,8,7,7,8,8,0,8

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

%C This is the 10's complement of A153042.

%C Equivalently, the 10-adic cube root of 1/9, i.e., x such that 9*x^3 = 1 (mod 10^n) for all n. - _M. F. Hasler_, Jan 02 2019

%H Seiichi Manyama, <a href="/A225412/b225412.txt">Table of n, a(n) for n = 0..10000</a>

%F p = ...711529, q = A153042 = ...288471, p + q = 0. - _Seiichi Manyama_, Aug 04 2019

%F Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 9, b(n) = b(n-1) + 7 * (9 * b(n-1)^3 - 1) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n. - _Seiichi Manyama_, Aug 13 2019

%e        9^3 == -1      (mod 10).

%e       29^3 == -11     (mod 10^2).

%e      529^3 == -111    (mod 10^3).

%e     1529^3 == -1111   (mod 10^4).

%e    11529^3 == -11111  (mod 10^5).

%e   711529^3 == -111111 (mod 10^6).

%p op([1,3],padic:-rootp(9*x^3  -1,  10, 101)); # _Robert Israel_, Aug 04 2019

%o (PARI) n=0; for(i=1, 100, m=(8*(10^i-1)/9)+1; for(x=0, 9, if(((n+(x*10^(i-1)))^3)%(10^i)==m, n=n+(x*10^(i-1)); print1(x", "); break)))

%o (PARI) upto(N=100, m=1/3)=Vecrev(digits(lift(chinese(Mod((1/9+O(5^N))^m, 5^N), Mod((1/9+O(2^N))^m, 2^N)))), N) \\ Following _Andrew Howroyd_'s code for A319740. - _M. F. Hasler_, Jan 02 2019

%o (PARI) Vecrev(digits(truncate(-(-1/9+O(10^100))^(1/3)))) \\ _Seiichi Manyama_, Aug 04 2019

%o (Ruby)

%o def A225412(n)

%o   ary = [9]

%o   a = 9

%o   n.times{|i|

%o     b = (a + 7 * (9 * a ** 3 - 1)) % (10 ** (i + 2))

%o     ary << (b - a) / (10 ** (i + 1))

%o     a = b

%o   }

%o   ary

%o end

%o p A225412(100) # _Seiichi Manyama_, Aug 13 2019

%Y Cf. A309600, A319740 (10-adic cube root of 1/11).

%Y Digits of 10-adic integers:

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

%Y A225406 (     9^(1/3));

%Y A225409 (  (-9)^(1/3)).

%K nonn,base

%O 0,1

%A _Aswini Vaidyanathan_, May 07 2013

%E Name edited by _Seiichi Manyama_, Aug 04 2019