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A309600 Digits of the 10-adic integer (17/9)^(1/3).

%I A309600 #41 Aug 11 2019 11:18:41
%S A309600 7,1,6,8,7,0,3,3,3,6,5,2,7,8,7,2,6,7,1,1,0,3,3,2,4,5,6,5,3,6,5,3,3,3,
%T A309600 7,5,2,4,7,5,0,2,9,0,6,7,0,8,8,6,6,7,0,1,2,4,5,3,2,8,6,9,7,3,1,6,6,9,
%U A309600 5,0,1,6,4,6,8,0,3,8,5,9,6,1,3,5,3,7,9,7,2,3,6,6,9,0,0,0,5,3,7,7,2
%N A309600 Digits of the 10-adic integer (17/9)^(1/3).
%H A309600 Seiichi Manyama, <a href="/A309600/b309600.txt">Table of n, a(n) for n = 0..10000</a>
%F A309600 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + 3 * (9 * b(n-1)^3 - 17) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.
%e A309600       7^3 == 3      (mod 10).
%e A309600      17^3 == 13     (mod 10^2).
%e A309600     617^3 == 113    (mod 10^3).
%e A309600    8617^3 == 1113   (mod 10^4).
%e A309600   78617^3 == 11113  (mod 10^5).
%e A309600   78617^3 == 111113 (mod 10^6).
%o A309600 (PARI) N=100; Vecrev(digits(lift(chinese(Mod((17/9+O(2^N))^(1/3), 2^N), Mod((17/9+O(5^N))^(1/3), 5^N)))), N)
%o A309600 (Ruby)
%o A309600 def A309600(n)
%o A309600   ary = [7]
%o A309600   a = 7
%o A309600   n.times{|i|
%o A309600     b = (a + 3 * (9 * a ** 3 - 17)) % (10 ** (i + 2))
%o A309600     ary << (b - a) / (10 ** (i + 1))
%o A309600     a = b
%o A309600   }
%o A309600   ary
%o A309600 end
%o A309600 p A309600(100)
%Y A309600 10-adic integer x.
%Y A309600 A225404       (x^3 = ...000003).
%Y A309600 A225405       (x^3 = ...000007).
%Y A309600 A225406       (x^3 = ...000009).
%Y A309600 A153042       (x^3 = ...111111).
%Y A309600 this sequence (x^3 = ...111113).
%Y A309600 A309601       (x^3 = ...111117).
%Y A309600 A309602       (x^3 = ...111119).
%Y A309600 A309603       (x^3 = ...222221).
%Y A309600 A225410       (x^3 = ...222223).
%Y A309600 A309604       (x^3 = ...222227).
%Y A309600 A309605       (x^3 = ...222229).
%Y A309600 A309606       (x^3 = ...333331).
%Y A309600 A225402       (x^3 = ...333333).
%Y A309600 A309569       (x^3 = ...333337).
%Y A309600 A309570       (x^3 = ...333339).
%Y A309600 A309595       (x^3 = ...444441).
%Y A309600 A309608       (x^3 = ...444443).
%Y A309600 A309609       (x^3 = ...444447).
%Y A309600 A309610       (x^3 = ...444449).
%Y A309600 A309611       (x^3 = ...555551).
%Y A309600 A309612       (x^3 = ...555553).
%Y A309600 A309613       (x^3 = ...555557).
%Y A309600 A309614       (x^3 = ...555559).
%Y A309600 A309640       (x^3 = ...666661).
%Y A309600 A309641       (x^3 = ...666663).
%Y A309600 A225411       (x^3 = ...666667).
%Y A309600 A309642       (x^3 = ...666669).
%Y A309600 A309643       (x^3 = ...777771).
%Y A309600 A309644       (x^3 = ...777773).
%Y A309600 A225401       (x^3 = ...777777).
%Y A309600 A309645       (x^3 = ...777779).
%Y A309600 A309646       (x^3 = ...888881).
%Y A309600 A309647       (x^3 = ...888883).
%Y A309600 A309648       (x^3 = ...888887).
%Y A309600 A225412       (x^3 = ...888889).
%Y A309600 A225409       (x^3 = ...999991).
%Y A309600 A225408       (x^3 = ...999993).
%Y A309600 A225407       (x^3 = ...999997).
%K A309600 nonn,base
%O A309600 0,1
%A A309600 _Seiichi Manyama_, Aug 09 2019