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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. 26
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; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 7th term in the sequence but 10 + 7 = 17 is not a term of the sequence). - Benoit Cloitre, Jun 18 2002

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

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.

(End)

For n > 1: A035612(a(n)) > 1. - Reinhard Zumkeller, Feb 03 2015

REFERENCES

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

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Joerg Arndt, Matters Computational (The Fxtbook)

M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Classic Sequences

Jiemeng Zhang, Zhixiong Wen, Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), #P2.52.

FORMULA

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

a(n) = A003622(n) - n. - Philippe Deléham, May 03 2004

a(n+1) = A022290(2*A003714(n)). - R. J. Mathar, Jan 31 2015

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

EXAMPLE

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

MAPLE

A022342 := proc(n)

      local g;

      g := (1+sqrt(5))/2 ;

    floor(n*g)-1 ;

end proc: # R. J. Mathar, Aug 04 2013

MATHEMATICA

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

PROG

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

(PARI) a(n)=(sqrtint(5*n^2)+n-2)\2 \\ Charles R Greathouse IV, Feb 27 2014

(Haskell)

a022342 n = a022342_list !! (n-1)

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

-- Reinhard Zumkeller, Mar 10 2013

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

CROSSREFS

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

Cf. A035612.

Sequence in context: A184588 A047488 A066093 * A218606 A077164 A062132

Adjacent sequences:  A022339 A022340 A022341 * A022343 A022344 A022345

KEYWORD

nonn,nice,easy

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified December 16 01:36 EST 2017. Contains 296063 sequences.