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Search: id:A022342
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| A022342 |
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Integers with "even" Zeckendorf expansions (do not end with ...+F1 = ...+1) (the Fibonacci-even numbers); also, apart from first term, a(n) = Fibonacci successor to n-1. |
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17
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| 0, 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The Zeckendorf expansion of n is obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains, for example 100 = 89 + 8 + 3.
The Fibonacci successor to n is found by replacing each F_i in the Zeckendorf expansion by F_{i+1}, for example the successor to 100 is 144 + 13 + 5 = 162.
If n appears n + (rank of n) does not (10 is the 7-th term in the sequence but 10 + 7 = 17 is not a term of the sequence) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 18 2002
Comments from Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30, 2001: "a(n)= Sum_{k\in A_n} F_{k+1}, where a(n)= Sum_{k\in A_n} F_k is the (unique) expression of n as a sum of ``noncontiguous'' Fibonacci numbers (with index >=2).
"a(10^n) gives the first few digits of g=(sqrt(5)+1)/2.
"The sequences given by b(n+1)=a(b(n)) obey the general recursion law of Fibonacci numbers. In particular the (sub)sequence (of a(-)) yielded by a starting value of 2=a(1), is the sequence of Fibonacci numbers >=2. Starting points of all such subsequences are given by A035336.
"a(n)=floor(phi*n+1/phi ); phi =(sqrt(5)+1)/2. a(F_n)=F_{n+1} if F_n is the N_th Fibonacci number."
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.
Zeckendorf, E., Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Joerg Arndt, Fxtbook
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
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FORMULA
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a(n) = [ n tau^2 ] - n - 1; or [ n tau ] -1.
a(n) = A003622(n) - n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 03 2004
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EXAMPLE
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The succesors to 1, 2, 3, 4=3+1 are 2, 3, 5, 7=5+2.
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PROGRAM
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(PARI) a(n)=floor(n*(sqrt(5)+1)/2)-1
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CROSSREFS
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Cf. A005206, A035336, A003622, A066096, A001950, A062879. Complement to A003622.
Adjacent sequences: A022339 A022340 A022341 this_sequence A022343 A022344 A022345
Sequence in context: A076798 A047488 A066093 this_sequence A077164 A062132 A003258
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Marc LeBrun (mlb(AT)well.com)
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