A241520 - OEIS

%I #17 Sep 08 2022 08:46:07

%S 34,55,123,144,212,233,301,322,390,411,479,500,568,589,657,678,746,

%T 767,835,856,924,945,1013,1034,1102,1123,1191,1212,1280,1301,1369,

%U 1390,1458,1479,1547,1568,1636,1657,1725,1746,1814,1835,1903,1924,1992,2013,2081

%N Numbers n such that n^2 == -1 (mod 89).

%C Numbers n such that n == 34 or 55 mod 89.

%H Vincenzo Librandi, <a href="/A241520/b241520.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F G.f.: x*(34 + 21*x + 34*x^2)/((1 + x)*(1 - x)^2).

%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.

%F a(n) = a(n-2) + 89 for all n>2.

%F a(n) = -34*(-1)^n + 89*floor(n/2).

%t 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 (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]] /* or */ [-34*(-1)^n+89*Floor(n/2): n in [1..50]];

%Y Cf. similar sequences listed in A155107.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Apr 25 2014