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A225402 10-adic cube root of -1/3.

%I A225402 #54 Apr 15 2022 14:27:04
%S A225402 7,7,4,6,4,4,5,8,3,4,4,6,8,3,1,5,1,0,5,1,6,9,6,1,3,1,3,7,2,2,4,6,4,4,
%T A225402 3,8,3,5,3,3,5,6,2,5,2,3,2,8,6,3,4,1,8,9,6,7,0,4,8,3,6,7,1,1,6,6,5,2,
%U A225402 0,7,4,0,9,2,8,0,3,0,3,0,7,3,9,7,9,7,3,4,6,5,3,1,3,0,0,8,7,4,5,7
%N A225402 10-adic cube root of -1/3.
%C A225402 Previous name was: 10-adic integer x such that x^3 == (10^(n+1)-1)/3 mod 10^(n+1) for n >= 0.
%C A225402 The 10-adic cube root of -1/3, i.e., x such that 3*x^3 = -1 (mod 10^n) for all n. - _M. F. Hasler_, Jan 02 2019
%H A225402 Seiichi Manyama, <a href="/A225402/b225402.txt">Table of n, a(n) for n = 0..10000</a>
%F A225402 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + 9 * (3 * b(n-1)^3 + 1) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n. - _Seiichi Manyama_, Aug 12 2019
%e A225402        7^3 == 3      (mod 10).
%e A225402       77^3 == 33     (mod 10^2).
%e A225402      477^3 == 333    (mod 10^3).
%e A225402     6477^3 == 3333   (mod 10^4).
%e A225402    46477^3 == 33333  (mod 10^5).
%e A225402   446477^3 == 333333 (mod 10^6).
%p A225402 b:= proc(n) option remember; `if`(n<2, 7*n,
%p A225402       irem(b(n-1)+9*(3*b(n-1)^3+1), 10^n))
%p A225402     end:
%p A225402 a:= n-> (b(n+1)-b(n))/10^n:
%p A225402 seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 15 2022
%o A225402 (PARI) n=0;for(i=1,100,m=(10^i-1)/3;for(x=0,9,if(((n+(x*10^(i-1)))^3)%(10^i)==m,n=n+(x*10^(i-1));print1(x", ");break)))
%o A225402 (PARI) upto(N=100, m=1/3)=Vecrev(digits(lift(chinese(Mod((-m+O(5^N))^m, 5^N), Mod((-m+O(2^N))^m, 2^N)))), N) \\ Following _Andrew Howroyd_'s code for A319740. - _M. F. Hasler_, Jan 02 2019
%o A225402 (Ruby)
%o A225402 def A225402(n)
%o A225402   ary = [7]
%o A225402   a = 7
%o A225402   n.times{|i|
%o A225402     b = (a + 9 * (3 * a ** 3 + 1)) % (10 ** (i + 2))
%o A225402     ary << (b - a) / (10 ** (i + 1))
%o A225402     a = b
%o A225402   }
%o A225402   ary
%o A225402 end
%o A225402 p A225402(100) # _Seiichi Manyama_, Aug 12 2019
%Y A225402 Cf. A225411, A309600, A319740 (10-adic cube root of 1/11).
%K A225402 nonn,base
%O A225402 0,1
%A A225402 _Aswini Vaidyanathan_, May 07 2013
%E A225402 New name (using _M. F. Hasler_'s comment) from _Joerg Arndt_, Aug 08 2019