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A290559 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 4 mod 7 (except for the initial 0).

%I A290559 #27 Dec 04 2022 12:39:40
%S A290559 0,4,39,235,235,12240,79468,667713,3961885,15491487,15491487,15491487,
%T A290559 7924798459,77131234464,561576286499,4630914723593,23621160763365,
%U A290559 189785813611370,1352938383547405,4609765579368303,4609765579368303,403571097067428308
%N A290559 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 4 mod 7 (except for the initial 0).
%C A290559 x   = ...450454,
%C A290559 x^2 = ...000002 = 2.
%H A290559 Seiichi Manyama, <a href="/A290559/b290559.txt">Table of n, a(n) for n = 0..1183</a>
%H A290559 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A290559 If n > 0, a(n) = 7^n - A290557(n).
%F A290559 a(0) = 0 and a(1) = 4, a(n) = a(n-1) + 6 * (a(n-1)^2 - 2) mod 7^n for n > 1.
%F A290559 a(n) == 2*T(7^n, 2) (mod 7^n) == (2 + sqrt(3))^(7^n) + (2 - sqrt(3))^(7^n) (mod 7^n), where T(n, x)  denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Dec 03 2022
%e A290559 a(1) = (    4)_7 = 4,
%e A290559 a(2) = (   54)_7 = 39,
%e A290559 a(3) = (  454)_7 = 235,
%e A290559 a(4) = (  454)_7 = 235,
%e A290559 a(5) = (50454)_7 = 12240.
%o A290559 (PARI) a(n) = if (n==0, 0, 7^n - truncate(sqrt(2+O(7^n)))); \\ _Michel Marcus_, Aug 06 2017
%Y A290559 Cf. A051277, A290557, A290558.
%K A290559 nonn,easy
%O A290559 0,2
%A A290559 _Seiichi Manyama_, Aug 05 2017