login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A089591 "Lazy binary" representation of n. Also called redundant binary representation of n. 5
0, 1, 10, 11, 20, 101, 110, 111, 120, 201, 210, 1011, 1020, 1101, 1110, 1111, 1120, 1201, 1210, 2011, 2020, 2101, 2110, 10111, 10120, 10201, 10210, 11011, 11020, 11101, 11110, 11111, 11120, 11201, 11210, 12011, 12020, 12101, 12110, 20111 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Let a(0) = 0 and construct a(n) from a(n-1) by (i) incrementing the rightmost digit and (ii) if any digit is 2, replace the rightmost 2 with a 0 and increment the digit immediately to its left. (Note that changing "if" to "while" in this recipe gives the standard binary representation of n, A007088(n)).
Equivalently, a(2n+1) = a(n):1 and a(2n+2) = b(n):0, where b(n) is obtained from a(n) by incrementing the least significant digit and : denotes string concatenation.
If the digits of a(n) are d_k, d_{k-1}, ..., d_2, d_1, d_0, then n = Sum_{i=0..k} d_i*2^i, just as in standard binary notation. The difference is that here we are a bit lazy, and allow a few digits to be 2's. The number of 2's in a(n) appears to be A037800(n+1). - N. J. A. Sloane, Jun 03 2023
Every pair of 2's is separated by a 0 and every pair of significant 0's is separated by a 2.
a(n) has exactly floor(log_2((n+2)/3))+1 digits [cf. A033484] and their sum is exactly floor(log_2(n+1)) [A000523].
The i-th digit of a(n) is ceiling( floor( ((n+1-2^i) mod 2^(i+1))/2^(i-1) ) / 2).
A137951 gives values of terms interpreted as ternary numbers, a(n)=A007089(A137951(n)). - Reinhard Zumkeller, Feb 25 2008
REFERENCES
Gerth S. Brodal, Worst-case efficient priority queues, SODA 1996.
Michael J. Clancy and D. E. Knuth, A programming and problem-solving seminar, Technical Report STAN-CS-77-606, Department of Computer Science, Stanford University, Palo Alto, 1977.
Haim Kaplan and Robert E. Tarjan, Purely functional representations of catenable sorted lists, STOC 1996.
Chris Okasaki, Purely Functional Data Structures, Cambridge, 1998.
LINKS
EXAMPLE
a(8) = 120 -> 121 -> 201 = a(9); a(9) = 201 -> 202 -> 210 = a(10).
MAPLE
A089591 := proc(n) option remember ; local nhalf ; if n <= 1 then RETURN(n) ; else nhalf := floor(n/2) ; if n mod 2 = 1 then RETURN(10*A089591(nhalf) +1) ; else RETURN(10*(A089591(nhalf-1)+1)) ; fi ; fi ; end: for n from 0 to 200 do printf("%d, ", A089591(n)) ; od ; # R. J. Mathar, Mar 11 2007
MATHEMATICA
a[n_] := a[n] = Module[{nhalf}, If[n <= 1, Return[n], nhalf = Floor[n/2]; If[Mod[n, 2]==1, Return[10*a[nhalf]+1], Return[10*(a[nhalf-1]+1)]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 19 2016, after R. J. Mathar *)
CROSSREFS
A158582: lazy binary different from regular binary, A089633: lazy binary and regular binary agree.
Sequence in context: A178361 A014418 A317204 * A064039 A255536 A298849
KEYWORD
easy,nonn,nice,base
AUTHOR
Jeff Erickson, Dec 29 2003
EXTENSIONS
More terms from R. J. Mathar, Mar 11 2007
Edited by Charles R Greathouse IV, Apr 30 2010
Edited by N. J. A. Sloane, Jun 03 2023
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 17 11:20 EDT 2024. Contains 371763 sequences. (Running on oeis4.)