A022342 - OEIS

%I #138 Jun 03 2024 14:27:11

%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 ...+F_2 = ...+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 k appears, k + (rank of k) 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 From _Amiram Eldar_, Sep 03 2022: (Start)

%C Numbers with an even number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is even.

%C The asymptotic density of this sequence is 1/phi (A094214). (End)

%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, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

%H Amiram Eldar, <a href="/A022342/b022342.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

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

%H A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See pages 3, 6.

%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 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%H Jiemeng Zhang, Zhixiong Wen, and Wen Wu, <a href="https://doi.org/10.37236/6745">Some Properties of the Fibonacci Sequence on an Infinite Alphabet</a>, Electronic Journal of Combinatorics, 24(2) (2017), Article 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 For n > 1: A035612(a(n)) > 1. - _Reinhard Zumkeller_, Feb 03 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

%F a(n) = a(n-1) + 1 + A005614(n-2) for n > 1; also  a(n) = a(n-1) + A014675(n-2) = a(n-1) + A001468(n-1). - _A.H.M. Smeets_, Apr 26 2024

%e The successors 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

%o (Python)

%o from math import isqrt

%o def A022342(n): return (n+isqrt(5*n**2)>>1)-1 # _Chai Wah Wu_, Aug 17 2022

%Y Positions of 0's in A003849.

%Y Complement of A003622.

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

%Y Cf. A035612, A094214, A104326, A356749.

%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - _N. J. A. Sloane_, Mar 11 2021

%K nonn,nice,easy

%O 1,2

%A _Marc LeBrun_

%E Name edited by _Peter Munn_, Dec 07 2021