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A204766 a(n) = 167*(n-1)-a(n-1) with n>1, a(1)=13.

%I A204766 #48 Apr 17 2025 14:55:05
%S A204766 13,154,180,321,347,488,514,655,681,822,848,989,1015,1156,1182,1323,
%T A204766 1349,1490,1516,1657,1683,1824,1850,1991,2017,2158,2184,2325,2351,
%U A204766 2492,2518,2659,2685,2826,2852,2993,3019,3160
%N A204766 a(n) = 167*(n-1)-a(n-1) with n>1, a(1)=13.
%C A204766 Positive numbers k such that k^2 == 2 (mod 167), where the prime 167 == -1 (mod 8).
%C A204766 Equivalently, numbers k such that k == 13 or 154 (mod 167).
%H A204766 Vincenzo Librandi, <a href="/A204766/b204766.txt">Table of n, a(n) for n = 1..1000</a>
%H A204766 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A204766 G.f.: x*(13+141*x+13*x^2)/((1+x)*(x-1)^2).
%F A204766 a(n) = (-167+115*(-1)^n+334*n)/4.
%F A204766 a(n) = a(n-1)+a(n-2)-a(n-3).
%F A204766 Sum_{n>=1} (-1)^(n+1)/a(n) = cot(13*Pi/167)*Pi/167. - _Amiram Eldar_, Feb 28 2023
%t A204766 CoefficientList[Series[x*(13+141*x+13*x^2)/((1+x)*(x-1)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 1, -1}, {13, 154, 180}, 40]
%o A204766 (Magma) [(-167+115*(-1)^n+334*n)/4: n in [1..60]];
%Y A204766 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.
%Y A204766 Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010, A206525, A206526.
%K A204766 nonn,easy
%O A204766 1,1
%A A204766 _Vincenzo Librandi_, Mar 09 2012