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 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 R. Zumkeller, Table of n, a(n) for n = 0..10000 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 Cf. A000523, A033484, A072998, A024629, A089600, A089601, A089604, A037800. A158582: lazy binary different from regular binary, A089633: lazy binary and regular binary agree. Sequence in context: A178361 A014418 A317204 * A064039 A255536 A298849 Adjacent sequences: A089588 A089589 A089590 * A089592 A089593 A089594 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

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