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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. 30
 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, 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. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) 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: A047488 A295282 A066093 * A218606 A077164 A062132 Adjacent sequences:  A022339 A022340 A022341 * A022343 A022344 A022345 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified May 27 16:47 EDT 2020. Contains 334664 sequences. (Running on oeis4.)