cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A089601 Interleaving of A089591 and A089600.

Original entry on oeis.org

0, 0, 1, 1, 2, 10, 11, 11, 12, 20, 21, 101, 102, 110, 111, 111, 112, 120, 121, 201, 202, 210, 211, 1011, 1012, 1020, 1021, 1101, 1102, 1110, 1111, 1111, 1112, 1120, 1121, 1201, 1202, 1210, 1211, 2011, 2012, 2020, 2021, 2101, 2102, 2110, 2111, 10111
Offset: 0

Views

Author

Jeff Erickson, Dec 31 2003

Keywords

Crossrefs

Cf. A089591, A089600.

Programs

  • 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: A089600 := proc(n) option remember ; local nhalf ; nhalf := floor(n/2) ; if n <= 1 then RETURN(n) ; else if n mod 2 = 1 then RETURN(10*A089600(nhalf) +1) ; else RETURN(10*A089591(nhalf-1)+2) ; fi ; fi ; end: A089601 := proc(n) if n mod 2 = 0 then A089600(n/2) ; else A089591((n-1)/2) ; fi ; end: seq(A089601(n),n=0..80) ; # R. J. Mathar, Jul 20 2007

Formula

To get a(n), drop last digit from either A089591(n) or A089600(n+1).

Extensions

More terms from R. J. Mathar, Jul 20 2007

A089591 "Lazy binary" representation of n. Also called redundant binary representation of n.

Original entry on oeis.org

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
Offset: 0

Views

Author

Jeff Erickson, Dec 29 2003

Keywords

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

Examples

			a(8) = 120 -> 121 -> 201 = a(9); a(9) = 201 -> 202 -> 210 = a(10).

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

Crossrefs

Cf. A000523, A033484, A072998, A024629, A089600, A089601, A089604, A037800.
A158582: lazy binary different from regular binary, A089633: lazy binary and regular binary agree.

Programs

  • 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 *)

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

A089604 "3-lazy binary" representation of n: to increment, add one to the last digit, then "carry" the rightmost 3 (replace 03->11, 13->21, or 23->31).

Original entry on oeis.org

0, 1, 2, 11, 12, 21, 22, 31, 112, 121, 122, 131, 212, 213, 222, 231, 312, 321, 1122, 1131, 1212, 1221, 1222, 1231, 1312, 1321, 2122, 2131, 2212, 2221, 2222, 2231, 2312, 2321, 3122, 3131, 3212, 3221, 11222
Offset: 0

Views

Author

Jeff Erickson, Dec 31 2003

Keywords

Comments

Obvious generalization of A089591 and A089600.

Extensions

Edited by Charles R Greathouse IV, Aug 03 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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