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A290804 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).

%I A290804 #28 Dec 03 2022 07:51:18
%S A290804 0,5,12,306,306,2707,69935,658180,4775895,10540696,10540696,575491194,
%T A290804 4530144680,59895293484,544340345519,1900786491217,20891032530989,
%U A290804 87356893670191,319987407657398,10090468995120092,44287154551239521,203871687146463523
%N A290804 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).
%C A290804 x   = ...410615,
%C A290804 x^2 = ...666664 = -3.
%H A290804 Seiichi Manyama, <a href="/A290804/b290804.txt">Table of n, a(n) for n = 0..1183</a>
%H A290804 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 A290804 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A290804 a(0) = 0 and a(1) = 5, a(n) = a(n-1) + 2 * (a(n-1)^2 + 3) mod 7^n for n > 1.
%F A290804 If n > 0, a(n) = 7^n - A290803(n).
%F A290804 a(n) = L(7^n,5) (mod 7^n) = ( ((5 + sqrt(29))/2)^(7^n) + ((5 - sqrt(29))/2)^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - _Peter Bala_, Nov 28 2022
%e A290804 a(1) =     5_7 = 5,
%e A290804 a(2) =    15_7 = 12,
%e A290804 a(3) =   615_7 = 306,
%e A290804 a(4) =   615_7 = 306,
%e A290804 a(5) = 10615_7 = 2707.
%o A290804 (PARI) a(n) = if (n, 7^n - truncate(sqrt(-3+O(7^(n)))), 0)
%Y A290804 Cf. A114525, A290797, A290803.
%K A290804 nonn,easy
%O A290804 0,2
%A A290804 _Seiichi Manyama_, Aug 11 2017