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A184327 a(1)=1, a(2)=17; thereafter a(n) = 6*a(n-1)-a(n-2)+c, where c=-4 if n is odd, c=12 if n is even.

%I A184327 #35 Sep 08 2022 08:45:55
%S A184327 1,17,97,577,3361,19601,114241,665857,3880897,22619537,131836321,
%T A184327 768398401,4478554081,26102926097,152139002497,886731088897,
%U A184327 5168247530881,30122754096401,175568277047521,1023286908188737,5964153172084897,34761632124320657
%N A184327 a(1)=1, a(2)=17; thereafter a(n) = 6*a(n-1)-a(n-2)+c, where c=-4 if n is odd, c=12 if n is even.
%H A184327 Vincenzo Librandi, <a href="/A184327/b184327.txt">Table of n, a(n) for n = 1..1000</a>
%H A184327 J. V. Leyendekkers and A. G. Shannon, <a href="http://nntdm.net/volume-18-2012/number-2/58-62/">Pellian sequence relationships among pi, e, sqrt(2)</a>, Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See Table 3, {y_n}.
%H A184327 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,0,-6,1).
%F A184327 From _Bruno Berselli_, Dec 26 2012: (Start)
%F A184327 G.f.: x*(1+11*x-5*x^2+x^3)/((1-x)*(1+x)*(1-6*x+x^2)).
%F A184327 a(n) = a(-n) = 6*a(n-1)-6*a(n-3)+a(n-4).
%F A184327 a(n) = ((1+sqrt(2))^(2n)+(1-sqrt(2))^(2n))/2+(-1)^n-1.
%F A184327 a(n) = 2*A090390(n)-1. (End)
%t A184327 CoefficientList[Series[(1 + 11 x - 5 x^2 + x^3)/((1 - x) (1 + x) (1 - 6 x + x^2)), {x, 0, 24}], x] (* _Bruno Berselli_, Dec 26 2012 *)
%o A184327 (Magma) /* By definition: */ a:=[1,17]; c:=func<n | IsOdd(n) select -4 else 12>; [n le 2 select a[n] else 6*Self(n-1)-Self(n-2)+c(n): n in [1..22]]; // _Bruno Berselli_, Dec 26 2012
%K A184327 nonn,easy
%O A184327 1,2
%A A184327 _N. J. A. Sloane_, Dec 23 2012
%E A184327 Edited from _Bruno Berselli_, Dec 26 2012