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%I #37 Mar 21 2021 13:00:34
%S 0,13,251,1985,56028,390112,390112,96940388,3379649772,24306922095,
%T 1565949316556,5597937117454,553948278039582,6380170650337192,
%U 154948841143926247,2848994066094341111,5711417117604156904,735629295252607184119,7353551390343301297535
%N One of the two successive approximations up to 17^n for 17-adic integer sqrt(-1). Here the 13 (mod 17) case (except for n=0).
%C x = ...04B6ED,
%C x^2 = ...GGGGGG = -1.
%H Seiichi Manyama, <a href="/A286878/b286878.txt">Table of n, a(n) for n = 0..812</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F If n > 0, a(n) = 17^n - A286877(n).
%F a(0) = 0 and a(1) = 13, a(n) = a(n-1) + 15 * (a(n-1)^2 + 1) mod 17^n for n > 1.
%e a(1) = ( D)_17 = 13,
%e a(2) = ( ED)_17 = 251,
%e a(3) = ( 6ED)_17 = 1985,
%e a(4) = (B6ED)_17 = 56028.
%o (Ruby)
%o def A(k, m, n)
%o ary = [0]
%o a, mod = k, m
%o n.times{
%o b = a % mod
%o ary << b
%o a = b ** m
%o mod *= m
%o }
%o ary
%o end
%o def A286878(n)
%o A(13, 17, n)
%o end
%o p A286878(100)
%o (Python)
%o def A(k, m, n):
%o ary=[0]
%o a, mod = k, m
%o for i in range(n):
%o b=a%mod
%o ary.append(b)
%o a=b**m
%o mod*=m
%o return ary
%o def a286878(n): return A(13, 17, n)
%o print(a286878(100)) # _Indranil Ghosh_, Aug 03 2017, after Ruby
%o (PARI) a(n) = if (n, 17^n-truncate(sqrt(-1+O(17^n))), 0); \\ _Michel Marcus_, Aug 04 2017
%Y The two successive approximations up to p^n for p-adic integer sqrt(-1): A048898 and A048899 (p=5), A286840 and A286841 (p=13), A286877 and this sequence (p=17).
%K nonn
%O 0,2
%A _Seiichi Manyama_, Aug 02 2017
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