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A022310 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=5.

%I A022310 #26 Feb 20 2017 06:18:36
%S A022310 0,5,6,12,19,32,52,85,138,224,363,588,952,1541,2494,4036,6531,10568,
%T A022310 17100,27669,44770,72440,117211,189652,306864,496517,803382,1299900,
%U A022310 2103283,3403184,5506468,8909653,14416122,23325776,37741899,61067676,98809576,159877253
%N A022310 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=5.
%H A022310 Vincenzo Librandi, <a href="/A022310/b022310.txt">Table of n, a(n) for n = 0..1000</a>
%H A022310 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A022310 From _R. J. Mathar_, Apr 07 2011: (Start)
%F A022310 G.f.: -x*(-5 + 4*x)/((x - 1)*(x^2 + x - 1)).
%F A022310 a(n) = A022096(n) - 1. (End)
%F A022310 a(n) = 6*F(n) + F(n-1) - 1, where F = A000045. - _Bruno Berselli_, Feb 20 2017
%F A022310 From _Colin Barker_, Feb 20 2017: (Start)
%F A022310 a(n) = -1 + (2^(-1-n)*((1-t)^n*(-11+t) + (1+t)^n*(11+t))) / t where t=sqrt(5).
%F A022310 a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
%t A022310 LinearRecurrence[{2, 0, -1}, {0, 5, 6}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)
%o A022310 (PARI) concat(0, Vec(x*(5-4*x) / ((1-x)*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Feb 20 2017
%Y A022310 Cf. A000045, A022096.
%K A022310 nonn,easy
%O A022310 0,2
%A A022310 _N. J. A. Sloane_