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%I #23 Aug 12 2017 03:14:06
%S 0,1,22,120,120,9724,26531,144180,144180,17438583,259560225,259560225,
%T 259560225,83307283431,180196293838,2893088585234,17135773115063,
%U 116834564823866,582095592798280,10352577180260974,55948157921753546,454909489409813551
%N One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-6). These are the numbers congruent to 1 mod 7 (except for the initial 0).
%C x = ...140231,
%C x^2 = ...666661 = -6.
%H Seiichi Manyama, <a href="/A290800/b290800.txt">Table of n, a(n) for n = 0..1183</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F a(0) = 0 and a(1) = 1, a(n) = a(n-1) + 3 * (a(n-1)^2 + 6) mod 7^n for n > 1.
%e a(1) = 1_7 = 1,
%e a(2) = 31_7 = 22,
%e a(3) = 231_7 = 120,
%e a(4) = 231_7 = 120,
%e a(5) = 40231_7 = 9724.
%p with(padic):
%p R:= [rootp(x^2+6,7,100)]:
%p R1:= op(select(t -> ratvaluep(evalp(t,7,1))=1, R)):
%p seq(ratvaluep(evalp(R1,7,n)),n=0..100); # _Robert Israel_, Aug 11 2017
%o (PARI) a(n) = if (n, truncate(sqrt(-6+O(7^(n)))), 0)
%Y Cf. A290794, A290802.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 10 2017
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