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A014417
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Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary words starting with 1 not containing 11, with the word 0 added.
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124
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0, 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100001, 100010, 100100, 100101, 101000, 101001, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001001, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101
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OFFSET
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0,3
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COMMENTS
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Old name was: Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary vectors not containing 11.
For n > 0, write n = Sum_{i >= 2} eps(i) Fib_i where eps(i) = 0 or 1 and no 2 consecutive eps(i) can be 1 (see A035517); then a(n) is obtained by writing the eps(i) in reverse order.
"One of the most important properties of the Fibonacci numbers is the special way in which they can be used to represent integers. Let's write j >> k <==> j >= k+2. Then every positive integer has a unique representation of the form n = F_k1 + F_k2 + ... + F_kr, where k1 >> k2 >> ... >> kr >> 0. (This is 'Zeckendorf's theorem.') ... We can always find such a representation by using a "greedy" approach, choosing F_k1 to be the largest Fibonacci number =< n, then choosing F_k2 to be the largest that is =< n - F_k1 and so on. Fibonacci representation needs a few more bits because adjacent 1's are not permitted; but the two representations are analogous." [Concrete Math.]
Since the binary representation of n in base of Fibonacci numbers allows only the successive bit pairs 00, 01, 10 and leaves 11 unused, we can use a ternary representation using all trits 0, 1, 2 where 00 --> 0, 01 --> 1 and 10 --> 2 (e.g. binary 1001010 as ternary 1022). - Daniel Forgues, Nov 30 2009
The same sequence also arises when considering the NegaFibonacci representations of the integers, as follows. Take the NegaFibonacci representations of n = 0, 1, 2, ... (A215022) and of n = -1, -2, -3, ... (A215023), sort the union of these two lists into increasing binary order, and we get A014417. Likewise the corresponding list of decimal representations, A003714, is the union of A215024 and A215025 sorted into increasing order. - N. J. A. Sloane, Aug 10 2012
The sequence of final bits, starting with a(1), is the complement of the Fibonacci word A005614. - N. J. A. Sloane, Oct 03 2018
This representation is named after the Belgian Army doctor and mathematician Edouard Zeckendorf (1901-1983). - Amiram Eldar, Jun 11 2021
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REFERENCES
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Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 169.
Edouard 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.
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LINKS
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Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished. See Table 2.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date. [Cached copy, with permission]
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EXAMPLE
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The Zeckendorf expansions of 1, 2, ... are 1 = 1 = Fib_2 -> 1, 2 = 2 = Fib_3 -> 10, 3 = Fib_4 -> 100, 4 = 3+1 = Fib_4 + Fib_2 -> 101, 5 = 5 = Fib_5 -> 1000, 6 = 1+5 = Fib_2 + Fib_5 -> 1001, etc.
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MAPLE
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local nshi, Z, i ;
if n <= 1 then
return n;
end if;
nshi := n ;
Z := [] ;
for i from A130234(n) to 2 by -1 do
if nshi >= A000045(i) and nshi > 0 then
Z := [1, op(Z)] ;
else
Z := [0, op(Z)] ;
end if;
end do:
add( op(i, Z)*10^(i-1), i=1..nops(Z)) ;
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MATHEMATICA
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fb[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n * Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; Table[ fb[n], {n, 0, 30}] (* Robert G. Wilson v, May 15 2004 *)
r = Map[Fibonacci, Range[2, 12]]; Table[Total[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] & /@ Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 0, 33}] (* Michael De Vlieger, Mar 27 2016, Version 10 *)
FromDigits/@Select[Tuples[{0, 1}, 7], SequenceCount[#, {1, 1}]==0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 14 2019 *)
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PROG
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(PARI) Zeckendorf(n)=my(k=0, v, m); while(fibonacci(k)<=n, k=k+1); m=k-1; v=vector(m-1); v[1]=1; n=n-fibonacci(k-1); while(n>0, k=0; while(fibonacci(k)<=n, k=k+1); v[m-k+2]=1; n=n-fibonacci(k-1)); v \\ Ralf Stephan
(PARI) Zeckendorf(n)= { local(k); a=0; while(n>0, k=0; while(fibonacci(k)<=n, k=k+1); a=a+10^(k-3); n=n-fibonacci(k-1); ); a }
{ for (n=0, 10000, Zeckendorf(n); print(n, " ", a); write("b014417.txt", n, " ", a) ) } \\ Harry J. Smith, Jan 17 2009
(Haskell)
a014417 0 = 0
a014417 n = foldl (\v z -> v * 10 + z) 0 $ a189920_row n
(Python)
from sympy import fibonacci
def a(n):
k=0
x=0
while n>0:
k=0
while fibonacci(k)<=n: k+=1
x+=10**(k - 3)
n-=fibonacci(k - 1)
return x
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CROSSREFS
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a(n) = A003714(n) converted to binary.
See A104326 for dual Zeckendorf representation of n.
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KEYWORD
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nonn,easy,base,nice
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AUTHOR
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EXTENSIONS
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Definition expanded and Duchene et al. reference added by N. J. A. Sloane, Aug 07 2018
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STATUS
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approved
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