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A290803 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).

%I A290803 #33 Jan 08 2025 11:06:00
%S A290803 0,2,37,37,2095,14100,47714,165363,988906,29812911,271934553,
%T A290803 1401835549,9311142521,36993716923,133882727330,2846775018726,
%U A290803 12341898038612,145273620317016,1308426190253051,1308426190253051,35505111746372480,354674176936820484
%N A290803 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 A290803 x   = ...256052,
%C A290803 x^2 = ...666664 = -3.
%H A290803 Seiichi Manyama, <a href="/A290803/b290803.txt">Table of n, a(n) for n = 0..1184</a>
%H A290803 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 A290803 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A290803 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 A290803 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 A290803 a(1) =     2_7 = 2,
%e A290803 a(2) =    52_7 = 37,
%e A290803 a(3) =    52_7 = 37,
%e A290803 a(4) =  6052_7 = 2095,
%e A290803 a(5) = 56052_7 = 14100.
%o A290803 (PARI) a(n) = if (n, truncate(sqrt(-3+O(7^(n)))), 0)
%Y A290803 Cf. A114525, A290796, A290804.
%K A290803 nonn,easy
%O A290803 0,2
%A A290803 _Seiichi Manyama_, Aug 11 2017