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A164581 a(n) = 5*a(n - 1) + a(n - 2), with a(0)=1, a(1)=2.

%I A164581 #35 Jul 13 2025 19:16:09
%S A164581 1,2,11,57,296,1537,7981,41442,215191,1117397,5802176,30128277,
%T A164581 156443561,812346082,4218173971,21903215937,113734253656,590574484217,
%U A164581 3066606674741,15923607857922,82684645964351,429346837679677,2229418834362736,11576441009493357
%N A164581 a(n) = 5*a(n - 1) + a(n - 2), with a(0)=1, a(1)=2.
%H A164581 Vincenzo Librandi, <a href="/A164581/b164581.txt">Table of n, a(n) for n = 0..1000</a>
%H A164581 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,1).
%F A164581 a(n) = 5*a(n-1)+a(n-2) = A052918(n)-3*A052918(n-1).
%F A164581 G.f.: (1-3*x)/(1-5*x-x^2).
%F A164581 a(n) = A052918(n) + A015449(n). - _R. J. Mathar_, Jul 06 2012
%F A164581 a(n) = (2^(-1-n)*((5-sqrt(29))^n*(1+sqrt(29))+(-1+sqrt(29))*(5+sqrt(29))^n))/sqrt(29). - _Colin Barker_, Oct 13 2015
%F A164581 a(n) = Sum_{k=0..n-2} A168561(n-2,k)*5^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*5^k, n>0. - _R. J. Mathar_, Feb 14 2024
%F A164581 a(n) = A052918(n) -3*A052918(n-1). - _R. J. Mathar_, Feb 14 2024
%F A164581 From _Peter Bala_, Jul 08 2025: (Start)
%F A164581 The following series telescope:
%F A164581 Sum_{n >= 1} 1/(a(n) - 7*(-1)^n/a(n)) = 3/10, since 1/(a(n) - 7*(-1)^n/a(n)) = b(n) - b(n+1), where b(n) = (1/5) * (a(n) + a(n-1)) / (a(n)*a(n-1)).
%F A164581 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 7*(-1)^n/a(n)) = 1/10, since 1/(a(n) - 7*(-1)^n/a(n)) = c(n) + c(n+1), where c(n) = (1/5) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)
%t A164581 LinearRecurrence[{5, 1}, {1, 2}, 40] (* or *) Rest[CoefficientList[Series [x (1 - 3 x) / (1 - 5 x - x^2), {x, 0, 40}], x]] (* _Harvey P. Dale_, May 02 2011 *)
%o A164581 (Magma) [ n le 2 select (n) else 5*Self(n-1)+Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Sep 12 2013
%o A164581 (PARI) Vec((1-3*x)/(1-5*x-x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015
%Y A164581 Cf. A000032, A001077, A052924, A078343, A179237, A180148, A329723.
%K A164581 nonn,easy
%O A164581 0,2
%A A164581 _Vincenzo Librandi_, Aug 17 2009