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Numbers n such that n^2 == -1 (mod 61).
2

%I #16 Sep 21 2024 15:12:55

%S 11,50,72,111,133,172,194,233,255,294,316,355,377,416,438,477,499,538,

%T 560,599,621,660,682,721,743,782,804,843,865,904,926,965,987,1026,

%U 1048,1087,1109,1148,1170,1209,1231,1270,1292,1331,1353,1392,1414,1453,1475,1514

%N Numbers n such that n^2 == -1 (mod 61).

%C Numbers n such that n == 11 or 50 (mod 61).

%H Vincenzo Librandi, <a href="/A241406/b241406.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*(11 + 39*x + 11*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) + 61 for all n>2.

%F a(n) = -11*(-1)^n + 61*floor(n/2).

%t Select[Range[1500], PowerMod[#, 2, 61] == 60 &] (* or *) CoefficientList[Series[(11 + 39 x + 11 x^2)/((1 + x) (1 - x)^2), {x, 0, 100}], x]

%o (Magma) I:=[11,50,72]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]];

%o (Magma) [-11*(-1)^n+61*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