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A241520 Numbers k such that k^2 == -1 (mod 89).

%I A241520 #23 Mar 12 2025 13:10:10
%S A241520 34,55,123,144,212,233,301,322,390,411,479,500,568,589,657,678,746,
%T A241520 767,835,856,924,945,1013,1034,1102,1123,1191,1212,1280,1301,1369,
%U A241520 1390,1458,1479,1547,1568,1636,1657,1725,1746,1814,1835,1903,1924,1992,2013,2081
%N A241520 Numbers k such that k^2 == -1 (mod 89).
%C A241520 Numbers k such that k == 34 or 55 (mod 89).
%H A241520 Vincenzo Librandi, <a href="/A241520/b241520.txt">Table of n, a(n) for n = 1..1000</a>
%H A241520 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A241520 G.f.: x*(34 + 21*x + 34*x^2)/((1 + x)*(1 - x)^2).
%F A241520 a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
%F A241520 a(n) = a(n-2) + 89 for all n>2.
%F A241520 a(n) = -34*(-1)^n + 89*floor(n/2).
%t A241520 Select[Range[1500], PowerMod[#, 2, 89] == 88 &] (* or *) CoefficientList[Series[(34 + 21 x + 34 x^2)/((1 + x) (1 -x)^2), {x, 0, 100}], x]
%o A241520 (Magma) I:=[34,55,123]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]];
%o A241520 (Magma) [-34*(-1)^n+89*Floor(n/2): n in [1..50]];
%Y A241520 Cf. similar sequences listed in A155107.
%K A241520 nonn,easy
%O A241520 1,1
%A A241520 _Vincenzo Librandi_, Apr 25 2014