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A182189 a(n) = 6*a(n-1) - a(n-2) - 4 with n > 1, a(0)=1, a(1)=3.

%I A182189 #77 Aug 16 2025 17:17:38
%S A182189 1,3,13,71,409,2379,13861,80783,470833,2744211,15994429,93222359,
%T A182189 543339721,3166815963,18457556053,107578520351,627013566049,
%U A182189 3654502875939,21300003689581,124145519261543,723573111879673,4217293152016491,24580185800219269,143263821649299119
%N A182189 a(n) = 6*a(n-1) - a(n-2) - 4 with n > 1, a(0)=1, a(1)=3.
%C A182189 If p is a prime of the form 8*n +- 1 then a(p) == 3 (mod p); if p is a prime of the form 8*n +- 3 then a(p) == -1 (mod p).
%C A182189 The terms a(n) > 1 satisfy a(n)^5 + b(n)^5 = c(n)^5 + d(n)^5 where b(n) = a(n) - 2, c(n) = (a(n)-1) + i*ceiling((a(n)-1)*sqrt(2)), and d(n) is the conjugate of c(n), where i is the imaginary unit. Note that Re(c(n)) is A001542(n) and Im(d(n)) is A001541(n). - _Pedro Caceres_, Dec 30 2017
%H A182189 Vincenzo Librandi, <a href="/A182189/b182189.txt">Table of n, a(n) for n = 0..1000</a>
%H A182189 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182189 G.f.: (1-4*x-x^2)/((1-x)*(1-6*x+x^2)). - _Bruno Berselli_, Jun 07 2012
%F A182189 a(n) = 1 + A000129(2*n). - _G. C. Greubel_, May 24 2021
%t A182189 m = -11;n = -1; c = 0;
%t A182189 list3 = Reap[While[c < 22, t = 6 n - m - 4; Sow[t];m = n; n = t;c++]][[2,1]]
%t A182189 CoefficientList[Series[(1-4*x-x^2)/((1-x)*(1-6*x+x^2)),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 26 2012 *)
%t A182189 1 + Fibonacci[2*Range[0, 40], 2] (* _G. C. Greubel_, May 24 2021 *)
%o A182189 (Magma) I:=[1,3]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)-4: n in [1..41]]; // _Bruno Berselli_, Jun 07 2012
%o A182189 (PARI) my(x='x+O('x^40)); Vec((1-4*x-x^2)/((1-x)*(1-6*x+x^2))) \\ _Altug Alkan_, Dec 30 2017
%o A182189 (Sage) [1 + lucas_number1(2*n,2,-1) for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y A182189 Cf. A000129, A001541, A001542, A182190, A182431.
%K A182189 nonn,easy
%O A182189 0,2
%A A182189 _Kenneth J Ramsey_, Apr 17 2012