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Search: id:A001611
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| 1, 2, 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n) = Fibonacci(n) + 1.
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
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.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
N. S. Mendelsohn, Permutations with restricted displacement, Canad. Math. Bull., 4 (1961), 29-38.
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LINKS
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INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 402
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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G.f.: (1-2*x^2)/(1-2*x+x^3).
a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
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.
a(n) = 2a(n-1) - a(n-3) - Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 13 2007
a(n)=(Fibonacci(n)+Fibonacci(n+3)+2)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
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MAPLE
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A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation.]
with(combinat): seq((fibonacci(n)+fibonacci(n+3)+2)/2, n=-2..35); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
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MATHEMATICA
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a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 2] + a[n - 1] - 1; Table[ a[n], {n, 0, 40} ]
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CROSSREFS
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Cf. A000045, A097280, A097281.
Sequence in context: A005856 A157876 A107293 this_sequence A039829 A143588 A032245
Adjacent sequences: A001608 A001609 A001610 this_sequence A001612 A001613 A001614
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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