login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.

REFERENCES

J L Brown 'A New Characterization of the Fibonacci Numbers' Fibonacci Quarterly, 3 (1965), pp. 1-8

LINKS

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

R 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 Zeck. rep. 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, 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, sequence extended - R. J. Mathar, Oct 23 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified July 31 17:29 EDT 2014. Contains 245085 sequences.