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A286877 One of the two successive approximations up to 17^n for 17-adic integer sqrt(-1). Here the 4 (mod 17) case (except for n=0).

%I A286877 #45 Jan 08 2025 11:03:04
%S A286877 0,4,38,2928,27493,1029745,23747457,313398285,3596107669,94280954402,
%T A286877 450044583893,28673959190179,28673959190179,3524407382568745,
%U A286877 13428985415474682,13428985415474682,42949774758062711577,91610966633729580058,6709533061724423693474
%N A286877 One of the two successive approximations up to 17^n for 17-adic integer sqrt(-1). Here the 4 (mod 17) case (except for n=0).
%C A286877 x   = ...GC5A24,
%C A286877 x^2 = ...GGGGGG = -1.
%H A286877 Seiichi Manyama, <a href="/A286877/b286877.txt">Table of n, a(n) for n = 0..813</a>
%H A286877 Peter Bala, <a href="/A210850/a210850.pdf">Using Lucas polynomials to find the p-adic square roots of -1, -2 and -3</a>, Dec 2022.
%H A286877 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A286877 a(0) = 0 and a(1) = 4, a(n) = a(n-1) + 2 * (a(n-1)^2 + 1) mod 17^n for n > 1.
%F A286877 a(n) == L(17^n,4) (mod 17^n) == (2 + sqrt(5))^(17^n) + (2 - sqrt(5))^(17^n) (mod 17^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - _Peter Bala_, Dec 02 2022
%e A286877 a(1) = (   4)_17 = 4,
%e A286877 a(2) = (  24)_17 = 38,
%e A286877 a(3) = ( A24)_17 = 2928,
%e A286877 a(4) = (5A24)_17 = 27493.
%o A286877 (Ruby)
%o A286877 def A(k, m, n)
%o A286877   ary = [0]
%o A286877   a, mod = k, m
%o A286877   n.times{
%o A286877     b = a % mod
%o A286877     ary << b
%o A286877     a = b ** m
%o A286877     mod *= m
%o A286877   }
%o A286877   ary
%o A286877 end
%o A286877 def A286877(n)
%o A286877   A(4, 17, n)
%o A286877 end
%o A286877 p A286877(100)
%o A286877 (Python)
%o A286877 def A(k, m, n):
%o A286877     ary=[0]
%o A286877     a, mod = k, m
%o A286877     for i in range(n):
%o A286877           b=a%mod
%o A286877           ary.append(b)
%o A286877           a=b**m
%o A286877           mod*=m
%o A286877     return ary
%o A286877 def a286877(n):
%o A286877     return A(4, 17, n)
%o A286877 print(a286877(100)) # _Indranil Ghosh_, Aug 03 2017
%o A286877 (PARI) a(n) = truncate(sqrt(-1+O(17^n))); \\ _Michel Marcus_, Aug 04 2017
%Y A286877 The two successive approximations up to p^n for p-adic integer sqrt(-1): A048898 and A048899 (p=5), A286840 and A286841 (p=13), this sequence and A286878 (p=17).
%K A286877 nonn,easy
%O A286877 0,2
%A A286877 _Seiichi Manyama_, Aug 02 2017