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