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A022319 a(n) = a(n-1) + a(n-2) + 1 for n > 1, a(0)=1, a(1)=5.

%I A022319 #37 Mar 09 2024 11:03:03
%S A022319 1,5,7,13,21,35,57,93,151,245,397,643,1041,1685,2727,4413,7141,11555,
%T A022319 18697,30253,48951,79205,128157,207363,335521,542885,878407,1421293,
%U A022319 2299701,3720995,6020697,9741693
%N A022319 a(n) = a(n-1) + a(n-2) + 1 for n > 1, a(0)=1, a(1)=5.
%H A022319 Reinhard Zumkeller, <a href="/A022319/b022319.txt">Table of n, a(n) for n = 0..1000</a>
%H A022319 Diana Savin and Elif Tan, <a href="https://arxiv.org/abs/2403.01592">On Companion sequences associated with Leonardo quaternions: Applications over finite fields</a>, arXiv:2403.01592 [math.CO], 2024. See p. 2.
%H A022319 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A022319 a(n) = Fibonacci(n-2) + Fibonacci(n+4) - 1. - _Zerinvary Lajos_, Feb 01 2008
%F A022319 From _R. J. Mathar_, Apr 07 2011: (Start)
%F A022319 G.f.: (1 + 3*x - 3*x^2) / ((1-x)*(1 - x - x^2)).
%F A022319 a(n) = A022112(n) - 1. (End)
%p A022319 with(combinat): seq(fibonacci(n-2)+fibonacci(n+4)-1, n=0..29); # _Zerinvary Lajos_, Feb 01 2008
%t A022319 LinearRecurrence[{2, 0, -1}, {1, 5, 7}, 40] (* _Harvey P. Dale_, Nov 19 2014 *)
%o A022319 (Haskell)
%o A022319 a022319 n = a022319_list !! (n-1)
%o A022319 a022319_list = 1 : 5 : zipWith (+)
%o A022319    (map (+ 1) a022319_list) (tail a022319_list)
%o A022319 -- _Reinhard Zumkeller_, May 26 2013
%o A022319 (PARI) x='x+O('x^50); Vec((1 +3*x -3*x^2)/((1-x)*(1 -x -x^2))) \\ _G. C. Greubel_, Aug 25 2017
%Y A022319 Cf. A192762 (partial sums).
%K A022319 nonn,easy
%O A022319 0,2
%A A022319 _N. J. A. Sloane_