%I #41 Aug 11 2019 11:18:41

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

%H Seiichi Manyama, <a href="/A309600/b309600.txt">Table of n, a(n) for n = 0..10000</a>

%F 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 7^3 == 3 (mod 10).

%e 17^3 == 13 (mod 10^2).

%e 617^3 == 113 (mod 10^3).

%e 8617^3 == 1113 (mod 10^4).

%e 78617^3 == 11113 (mod 10^5).

%e 78617^3 == 111113 (mod 10^6).

%o (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 (Ruby)

%o def A309600(n)

%o ary = [7]

%o a = 7

%o n.times{|i|

%o b = (a + 3 * (9 * a ** 3 - 17)) % (10 ** (i + 2))

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

%o a = b

%o }

%o ary

%o end

%o p A309600(100)

%Y 10-adic integer x.

%Y A225404 (x^3 = ...000003).

%Y A225405 (x^3 = ...000007).

%Y A225406 (x^3 = ...000009).

%Y A153042 (x^3 = ...111111).

%Y this sequence (x^3 = ...111113).

%Y A309601 (x^3 = ...111117).

%Y A309602 (x^3 = ...111119).

%Y A309603 (x^3 = ...222221).

%Y A225410 (x^3 = ...222223).

%Y A309604 (x^3 = ...222227).

%Y A309605 (x^3 = ...222229).

%Y A309606 (x^3 = ...333331).

%Y A225402 (x^3 = ...333333).

%Y A309569 (x^3 = ...333337).

%Y A309570 (x^3 = ...333339).

%Y A309595 (x^3 = ...444441).

%Y A309608 (x^3 = ...444443).

%Y A309609 (x^3 = ...444447).

%Y A309610 (x^3 = ...444449).

%Y A309611 (x^3 = ...555551).

%Y A309612 (x^3 = ...555553).

%Y A309613 (x^3 = ...555557).

%Y A309614 (x^3 = ...555559).

%Y A309640 (x^3 = ...666661).

%Y A309641 (x^3 = ...666663).

%Y A225411 (x^3 = ...666667).

%Y A309642 (x^3 = ...666669).

%Y A309643 (x^3 = ...777771).

%Y A309644 (x^3 = ...777773).

%Y A225401 (x^3 = ...777777).

%Y A309645 (x^3 = ...777779).

%Y A309646 (x^3 = ...888881).

%Y A309647 (x^3 = ...888883).

%Y A309648 (x^3 = ...888887).

%Y A225412 (x^3 = ...888889).

%Y A225409 (x^3 = ...999991).

%Y A225408 (x^3 = ...999993).

%Y A225407 (x^3 = ...999997).

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Aug 09 2019