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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. 28


%S 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,

%T 41,42,44,45,47,49,50,52,54,55,57,58,60,62,63,65,66,68,70,71,73,75,76,

%U 78,79,81,83,84,86,87,89,91,92,94,96,97,99,100,102,104,105,107

%N 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 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 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 If n appears, n + (rank of n) does not (10 is the 7th term in the sequence but 10 + 7 = 17 is not a term of the sequence). - _Benoit Cloitre_, Jun 18 2002

%C From Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001: (Start)

%C 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 a(10^n) gives the first few digits of g=(sqrt(5)+1)/2.

%C 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 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.

%C (End)

%C For n > 1: A035612(a(n)) > 1. - _Reinhard Zumkeller_, Feb 03 2015

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.

%D E. Zeckendorf, 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 T. D. Noe, <a href="/A022342/b022342.txt">Table of n, a(n) for n = 1..1000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>

%H M. Rigo, P. Salimov, and E. Vandomme, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Rigo/rigo3.html">Some Properties of Abelian Return Words</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>

%H Jiemeng Zhang, Zhixiong Wen, Wen Wu, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p52">Some Properties of the Fibonacci Sequence on an Infinite Alphabet</a>, Electronic Journal of Combinatorics, 24(2) (2017), #P2.52.

%F a(n) = floor(n*phi^2) - n - 1 = floor(n*phi) - 1 = A000201(n) - 1, where phi is the golden ratio.

%F a(n) = A003622(n) - n. - _Philippe Deléham_, May 03 2004

%F a(n+1) = A022290(2*A003714(n)). - _R. J. Mathar_, Jan 31 2015

%F a(n) = A000201(n)-1. First differences are given in A014675 (or A001468, ignoring its first term). - _M. F. Hasler_, Oct 13 2017

%e The succesors to 1, 2, 3, 4=3+1 are 2, 3, 5, 7=5+2.

%p A022342 := proc(n)

%p local g;

%p g := (1+sqrt(5))/2 ;

%p floor(n*g)-1 ;

%p end proc: # _R. J. Mathar_, Aug 04 2013

%t With[{t=GoldenRatio^2},Table[Floor[n*t]-n-1,{n,70}]] (* _Harvey P. Dale_, Aug 08 2012 *)

%o (PARI) a(n)=floor(n*(sqrt(5)+1)/2)-1

%o (PARI) a(n)=(sqrtint(5*n^2)+n-2)\2 \\ _Charles R Greathouse IV_, Feb 27 2014

%o (Haskell)

%o a022342 n = a022342_list !! (n-1)

%o a022342_list = filter ((notElem 1) . a035516_row) [0..]

%o -- _Reinhard Zumkeller_, Mar 10 2013

%o (MAGMA) [Floor(n*(Sqrt(5)+1)/2)-1: n in [1..100]]; // _Vincenzo Librandi_, Feb 16 2015

%Y Cf. A000201, A005206, A035336, A003622, A066096, A001950, A062879, A035516, A026274. Complement to A003622.

%Y Cf. A035612.

%K nonn,nice,easy

%O 1,2

%A _Marc LeBrun_

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Last modified February 18 02:20 EST 2018. Contains 299297 sequences. (Running on oeis4.)