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%I #31 Dec 03 2022 07:52:17
%S 0,2,37,37,2095,14100,47714,165363,988906,29812911,271934553,
%T 1401835549,9311142521,36993716923,133882727330,2846775018726,
%U 12341898038612,145273620317016,1308426190253051,1308426190253051,35505111746372480,354674176936820484
%N One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 2 mod 7 (except for the initial 0).
%C x = ...256052,
%C x^2 = ...666664 = -3.
%H Seiichi Manyama, <a href="/A290803/b290803.txt">Table of n, a(n) for n = 0..1184</a>
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F a(0) = 0 and a(1) = 2, a(n) = a(n-1) + 5 * (a(n-1)^2 + 3) mod 7^n for n > 1.
%F a(n) = L(7^n,2) (mod 7^n) = ( (1 + sqrt(2))^(7^n) + (1 - sqrt(2))^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - _Peter Bala_, Nov 28 2022
%e a(1) = 2_7 = 2,
%e a(2) = 52_7 = 37,
%e a(3) = 52_7 = 37,
%e a(4) = 6052_7 = 2095,
%e a(5) = 56052_7 = 14100.
%o (PARI) a(n) = if (n, truncate(sqrt(-3+O(7^(n)))), 0)
%Y Cf. A114525, A290796, A290804.
%K nonn,easy
%O 0,2
%A _Seiichi Manyama_, Aug 11 2017
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