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A290800 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).

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