%I A001611 M0288 N0103
%S A001611 1,2,2,3,4,6,9,14,22,35,56,90,145,234,378,611,988,1598,2585,4182,
%T A001611 6766,10947,17712,28658,46369,75026,121394,196419,317812,514230,
%U A001611 832041,1346270,2178310,3524579,5702888,9227466,14930353,24157818
%N A001611 Fibonacci numbers (A000045) + 1.
%C A001611 a(n) = Fibonacci(n) + 1.
%C A001611 a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible 
               with three successive terms as sides. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Jun 03 2003
%C A001611 a(n+2)=A^(n)B(1), n>=0, with compositions of Wythoff's complementary 
               A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link 
               under A135817 for the Wythoff representation of numbers (with A as 
               1 and B as 0 and the argument 1 omitted). E.g. 2=`0`, 3=`10`, 4=`110`, 
               6=`1110`,..., in Wythoff code.
%D A001611 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001611 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001611 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001611 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A001611 N. S. Mendelsohn, Permutations with restricted displacement, Canad. Math. 
               Bull., 4 (1961), 29-38.
%H A001611 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=402">
               Encyclopedia of Combinatorial Structures 402</a>
%H A001611 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001611 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A001611 G.f.: (1-2*x^2)/(1-2*x+x^3).
%F A001611 a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
%F A001611 Fibonacci(4n) + 1 = Fibonacci(2n-1)*Lucas(2n+1); Fibonacci(4n+1) + 1 
               = Fibonacci(2n+1)*Lucas(2n); Fibonacci(4n+2) + 1 = Fibonacci(2n+2)*Lucas(2n); 
               Fibonacci(4n+3) + 1 = Fibonacci(2n+1)*Lucas(2n+2). - R. K. Guy, Feb 
               27, 2003.
%F A001611 a(n) = 2a(n-1) - a(n-3) - Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 
               13 2007
%F A001611 a(n)=(Fibonacci(n)+Fibonacci(n+3)+2)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 01 2008
%p A001611 A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation.]
%p A001611 with(combinat): seq((fibonacci(n)+fibonacci(n+3)+2)/2, n=-2..35); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
%t A001611 a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 2] + a[n - 1] - 1; Table[ a[n], 
               {n, 0, 40} ]
%Y A001611 Cf. A000045, A097280, A097281.
%Y A001611 Sequence in context: A005856 A157876 A107293 this_sequence A039829 A143588 
               A032245
%Y A001611 Adjacent sequences: A001608 A001609 A001610 this_sequence A001612 A001613 
               A001614
%K A001611 nonn,easy
%O A001611 0,2
%A A001611 N. J. A. Sloane (njas(AT)research.att.com).