%I #31 Feb 28 2023 03:00:11
%S 51,62,164,175,277,288,390,401,503,514,616,627,729,740,842,853,955,
%T 966,1068,1079,1181,1192,1294,1305,1407,1418,1520,1531,1633,1644,1746,
%U 1757,1859,1870,1972,1983,2085,2096,2198,2209
%N a(n) = 113*(n-1) - a(n-1) with n>1, a(1)=51.
%C Positive numbers k such that k^2 == 2 (mod 113), where the prime 113 == 1 (mod 8).
%C Equivalently, numbers k such that k == 51 or 62 (mod 113).
%H Vincenzo Librandi, <a href="/A206525/b206525.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 a(n) = a(n-2) + 113.
%F G.f.: x*(51+11*x+51*x^2)/((1+x)*(x-1)^2).
%F a(n) = (-113-91*(-1)^n+226*n)/4.
%F a(n) = a(n-1)+a(n-2)-a(n-3).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = tan(11*Pi/226)*Pi/113. - _Amiram Eldar_, Feb 28 2023
%t LinearRecurrence[{1, 1, -1}, {51, 62, 164}, 40] (* or *) CoefficientList[Series[x*(51+11*x+51*x^2)/((1+x)*(x-1)^2), {x, 0, 40}], x] (* or *) a[1] = 51; a[n_] := a[n] = 113*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]
%o (Magma) [(-113-91*(-1)^n+226*n)/4: n in [1..60]];
%Y Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010.
%Y Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204769.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 09 2012