Search: id:A022342
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%I A022342
%S A022342 0,2,3,5,7,8,10,11,13,15,16,18,20,21,23,24,26,28,29,31,32,34,36,37,
%T A022342 39,41,42,44,45,47,49,50,52,54,55,57,58,60,62,63,65,66,68,70,71,73,
%U A022342 75,76,78,79,81,83,84,86,87,89,91,92,94,96,97,99,100,102,104,105,107
%N A022342 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.
%C A022342 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.
%C A022342 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.
%C A022342 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
%C A022342 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).
%C A022342 "a(10^n) gives the first few digits of g=(sqrt(5)+1)/2.
%C A022342 "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.
%C A022342 "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."
%D A022342 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 307-308 of 2nd edition.
%D A022342 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.
%H A022342 T. D. Noe, <a href="b022342.txt">Table of n, a(n) for n=1..1000</a>
%H A022342 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A022342 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A022342 N. J. A. Sloane, <a href="classic.html#WYTH">Classic Sequences</a>
%F A022342 a(n) = [ n tau^2 ] - n - 1; or [ n tau ] -1.
%F A022342 a(n) = A003622(n) - n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               May 03 2004
%e A022342 The succesors to 1, 2, 3, 4=3+1 are 2, 3, 5, 7=5+2.
%o A022342 (PARI) a(n)=floor(n*(sqrt(5)+1)/2)-1
%Y A022342 Cf. A005206, A035336, A003622, A066096, A001950, A062879. Complement 
               to A003622.
%Y A022342 Sequence in context: A076798 A047488 A066093 this_sequence A077164 A062132 
               A003258
%Y A022342 Adjacent sequences: A022339 A022340 A022341 this_sequence A022343 A022344 
               A022345
%K A022342 nonn,nice,easy
%O A022342 1,2
%A A022342 Marc LeBrun (mlb(AT)well.com)

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