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A290806 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-5). These are the numbers congruent to 3 mod 7 (except for the initial 0).

%I A290806 #21 Aug 14 2017 03:11:29
%S A290806 0,3,17,311,997,3398,20205,608450,2255536,25314740,25314740,307789989,
%T A290806 8217096961,77423532966,368090564187,4437429001281,4437429001281,
%U A290806 4437429001281,4437429001281,3261264624822179,3261264624822179,3261264624822179,1120352992791390193
%N A290806 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-5). These are the numbers congruent to 3 mod 7 (except for the initial 0).
%C A290806 x   = ...112623,
%C A290806 x^2 = ...666662 = -5.
%H A290806 Robert Israel, <a href="/A290806/b290806.txt">Table of n, a(n) for n = 0..1182</a>
%H A290806 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A290806 a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 + 5) mod 7^n for n > 1.
%e A290806 a(1) =    3_7 = 3,
%e A290806 a(2) =   23_7 = 17,
%e A290806 a(3) =  623_7 = 311,
%e A290806 a(4) = 2623_7 = 997.
%p A290806 with(padic):
%p A290806 R:= [rootp(x^2+5, 7, 100)]:
%p A290806 R1:= op(select(t -> ratvaluep(evalp(t, 7, 1))=3, R)):
%p A290806 seq(ratvaluep(evalp(R1, 7, n)), n=0..100); # _Robert Israel_, Aug 13 2017
%o A290806 (PARI) a(n) = if (n, truncate(sqrt(-5+O(7^(n)))), 0)
%Y A290806 Cf. A290798, A290809.
%K A290806 nonn
%O A290806 0,2
%A A290806 _Seiichi Manyama_, Aug 11 2017