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A022308 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=3.

%I A022308 #41 Mar 26 2018 11:45:48
%S A022308 0,3,4,8,13,22,36,59,96,156,253,410,664,1075,1740,2816,4557,7374,
%T A022308 11932,19307,31240,50548,81789,132338,214128,346467,560596,907064,
%U A022308 1467661,2374726,3842388,6217115,10059504,16276620,26336125,42612746,68948872,111561619
%N A022308 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=3.
%H A022308 Colin Barker, <a href="/A022308/b022308.txt">Table of n, a(n) for n = 0..1000</a>
%H A022308 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A022308 G.f.: x*(3-2*x) / (x^3-2*x+1).
%F A022308 a(n) = 2*A000045(n) + A000045(n+2) - 1 = A000285(n)-1.
%F A022308 a(n) = 2*a(n-1) - a(n-3) for n>=3. - _Ron Knott_, Aug 25 2006
%F A022308 a(n) = (3*A000032(n) - A000045(n) - 2)/2. - _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012
%F A022308 a(n) = 4*F(n) + F(n-1) - 1, where F = A000045. - _Bruno Berselli_, Feb 20 2017
%F A022308 a(n) = (-10 + (5-7*sqrt(5))*((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n*(5+7*sqrt(5))) / 10. - _Colin Barker_, Feb 20 2017
%p A022308 with(combinat): seq(fibonacci(n)+fibonacci(n+5)-1, n=-2..30); # _Zerinvary Lajos_, Feb 01 2008
%t A022308 Table[(3 LucasL[n] - Fibonacci[n] - 2)/2, {n, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%t A022308 LinearRecurrence[{2, 0, -1}, {0, 3, 4}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%t A022308 CoefficientList[Series[x (3 - 2 x)/(x^3 - 2 x + 1), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 26 2018 *)
%o A022308 (PARI) concat(0, Vec(x*(3-2*x)/(x^3-2*x+1) + O(x^50))) \\ _Colin Barker_, Feb 20 2017
%o A022308 (PARI) a(n) = if(n==0, 0, if(n==1, 3, a(n-1)+a(n-2)+1)) \\ _Felix Fröhlich_, Mar 26 2018
%Y A022308 Cf. A000032, A000045, A000285, A122195.
%K A022308 nonn,easy
%O A022308 0,2
%A A022308 _N. J. A. Sloane_