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%I M0288 N0103 #137 Sep 08 2022 08:44:29
%S 1,2,2,3,4,6,9,14,22,35,56,90,145,234,378,611,988,1598,2585,4182,6766,
%T 10947,17712,28658,46369,75026,121394,196419,317812,514230,832041,
%U 1346270,2178310,3524579,5702888,9227466,14930353,24157818,39088170,63245987,102334156
%N a(n) = Fibonacci(n) + 1.
%C a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - _Amarnath Murthy_, Jun 03 2003
%C 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.
%C The first-difference sequence is the Fibonacci sequence (A000045). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
%C 2 and 3 are the only primes in this sequence.
%C a(n) is the number of 1 X n nonogram puzzles which can be solved uniquely. See A242876 for puzzle definition. - _Lior Manor_, Jan 23 2022
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001611/b001611.txt">Table of n, a(n) for n = 0..250</a>
%H Andrei Asinowski, Cyril Banderier and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H K.-W. Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Chen/chen70.html">Greatest Common Divisors in Shifted Fibonacci Sequences</a>, J. Int. Seq. 14 (2011) # 11.4.7.
%H Massimiliano Fasi and Gian Maria Negri Porzio, <a href="http://eprints.maths.manchester.ac.uk/2709/">Determinants of Normalized Bohemian Upper Hessemberg Matrices</a>, University of Manchester (England, 2019).
%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Griffiths/gr48.html">On a Matrix Arising from a Family of Iterated Self-Compositions</a>, Journal of Integer Sequences, 18 (2015), #15.11.8.
%H R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H Fumio Hazama, <a href="http://dx.doi.org/10.1016/j.disc.2011.06.008">Spectra of graphs attached to the space of melodies</a>, Discr. Math., 311 (2011), 2368-2383. See Table 5.1.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=402">Encyclopedia of Combinatorial Structures 402</a>
%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
%H N. S. Mendelsohn, <a href="http://dx.doi.org/10.4153/CMB-1961-005-4">Permutations with confined displacement</a>, Canad. Math. Bull., 4 (1961), 29-38.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F G.f.: (1-2*x^2)/(1-2*x+x^3).
%F a(n) = 2*a(n-1) - a(n-3). - _Tanya Khovanova_, Jul 13 2007
%F a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
%F F(4*n) + 1 = F(2*n-1)*L(2*n+1); F(4*n+1) + 1 = F(2*n+1)*L(2*n); F(4*n+2) + 1 = F(2*n+2)*L(2*n); F(4*n+3) + 1 = F(2*n+1)*L(2*n+2) where F(n)=Fibonacci(n) and L(n)=Lucas(n). - _R. K. Guy_, Feb 27 2003
%F a(1) = 2; a(n+1)=floor(a(n)*(sqrt(5)+1)/2). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
%F a(n) = Sum_{k=0..n+1} Fibonacci(k-3). - _Ehren Metcalfe_, Apr 15 2019
%p A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); # _Simon Plouffe_ in his 1992 dissertation
%p with(combinat): seq((fibonacci(n)+1), n=0..35);
%t a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n-2] + a[n-1] - 1; Table[ a[n], {n, 0, 40} ]
%t Fibonacci[Range[0,50]]+1 (* _Harvey P. Dale_, Mar 23 2011 *)
%o (PARI) a(n)=fibonacci(n)+1 \\ _Charles R Greathouse IV_, Jul 25 2011
%o (Magma) [Fibonacci(n)+1: n in [1..37]]; // _Bruno Berselli_, Jul 26 2011
%o (Haskell)
%o a001611 = (+ 1) . a000045
%o a001611_list = 1 : 2 : map (subtract 1)
%o (zipWith (+) a001611_list $ tail a001611_list)
%o -- _Reinhard Zumkeller_, Jul 30 2013
%Y Cf. A000045, A097280, A097281.
%Y Cf. A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by _N. J. A. Sloane_, Jun 25 2010 in response to a comment from _Aviezri S. Fraenkel_]
%Y Cf. A002062, A160536, A212272, A242876.
%K nonn,easy,hear
%O 0,2
%A _N. J. A. Sloane_
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