%I #27 Dec 04 2022 12:39:40

%S 0,4,39,235,235,12240,79468,667713,3961885,15491487,15491487,15491487,

%T 7924798459,77131234464,561576286499,4630914723593,23621160763365,

%U 189785813611370,1352938383547405,4609765579368303,4609765579368303,403571097067428308

%N 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 x = ...450454,

%C x^2 = ...000002 = 2.

%H Seiichi Manyama, <a href="/A290559/b290559.txt">Table of n, a(n) for n = 0..1183</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.

%F If n > 0, a(n) = 7^n - A290557(n).

%F 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 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 a(1) = ( 4)_7 = 4,

%e a(2) = ( 54)_7 = 39,

%e a(3) = ( 454)_7 = 235,

%e a(4) = ( 454)_7 = 235,

%e a(5) = (50454)_7 = 12240.

%o (PARI) a(n) = if (n==0, 0, 7^n - truncate(sqrt(2+O(7^n)))); \\ _Michel Marcus_, Aug 06 2017

%Y Cf. A051277, A290557, A290558.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Aug 05 2017