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A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. 7
0, 1, 10, 11, 101, 110, 111, 1010, 1011, 1101, 1110, 1111, 10101, 10110, 10111, 11010, 11011, 11101, 11110, 11111, 101010, 101011, 101101, 101110, 101111, 110101, 110110, 110111, 111010, 111011, 111101, 111110, 111111, 1010101 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Whereas the Zeckendorf (binary) rep (A014417) has no consecutive 1's (no two consecutive Fibonacci numbers in a set whose sum is n), the Dual Zeckendorf Representation has no consecutive 0's. Also called the Maximal (Binary) Fibonacci Representation, the Zeckendorf rep. being the Minimal in terms of number of 1's in the binary representation.

Also known as the lazy Fibonacci representation of n. - Glen Whitney, Oct 21 2017

LINKS

Table of n, a(n) for n=0..33.

J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 1-8.

Ron Knott, Using Fibonacci Numbers to Represent Whole Numbers.

FORMULA

a(n) = A007088(A003754(n+1)).

EXAMPLE

As a sum of Fibonacci numbers (A000045) [using 1 at most once], 13 is 13=8+5=8+3+2.

The largest set here is 8+3+2 or, in base Fibonacci, 10110 so a(13)=10110(fib).

The Zeckendorf representation would be the smallest set or {13}=100000(fib).

MAPLE

dualzeckrep:=proc(n)local i, z; z:=zeckrep(n); i:=1; while i<=nops(z)-2 do if z[i]=1 and z[i+1]=0 and z[i+2]=0 then z[i]:=0; z[i+1]:=1; z[i+2]:=1; if i>3 then i:=i-2 fi else i:=i+1 fi od; if z[1]=0 then z:=subsop(1=NULL, z) fi; z end proc: seq(dualzeckrep(n), n=0..20) ;

CROSSREFS

Cf. A014417, A095791, A104324.

Sequence in context: A055611 A077813 A203075 * A205598 A037090 A171676

Adjacent sequences:  A104323 A104324 A104325 * A104327 A104328 A104329

KEYWORD

nonn

AUTHOR

Ron Knott, Mar 01 2005

EXTENSIONS

Index in formula corrected, missing parts of the maple code recovered, and sequence extended by R. J. Mathar, Oct 23 2010

STATUS

approved

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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.