A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Author

Antti Karttunen, May 11 2001

Keywords

• nonn
• base
• easy

Comments

"msb" = "most significant bit", A053644.

These encode lattice walks using steps (+1,+1) (= 1's in binary expansion) and (+1,-1) (= 0's in binary expansion) that start from the origin (0,0) and never "dive" under the "sea-level" y=0.

The number of such walks of length n (here: the terms of binary width n) is given by C(n,floor(n/2)) = A001405, which is based on the fact mentioned in Guy's article that the shallow diagonals of the Catalan triangle A009766 sum to A001405.

From Jason Kimberley, Feb 08 2013: (Start)

This sequence is a subsequence of A072601.

Define a map from this set onto the nonnegative integers as follows: set the output bit string to be empty, representing zero; process the input string from left to right; when 1 occurs, change the rightmost 0 in the output to 1; if there is no 0 in the output, prepend a 1; when 0 occurs in the input, change the rightmost 1 in the output to 1. The definition of this sequence ensures that we always have a 1 in the output when a 0 occurs in the input. We this map is onto by showing the restriction to the subset Asubsequence is onto. (End)

The binary representation of a(n) is the numeric representation of the left half of a symmetric balanced string of parentheses with "(" representing 1 and ")" representing 0 (see comments and examples in A001405). Some of the numbers in this sequence cannot be realized as the 1-0-pattern of the odd/even positions of 1's in any row n of A237048 that determines the parts and their widths in the symmetric representation of sigma(n), see A352696. - Hartmut F. W. Hoft, Mar 29 2022

Examples

			From _Hartmut F. W. Hoft_, Mar 29 2022: (Start)
The columns in the table are the numbers n, the base-2 representation of n, the left half of the symmetric balanced string of parentheses corresponding to n, validity of the nondiving property for n, and associated number a(n):
1   1      (      True    a(1)
2   10     ()     True    a(2)
3   11     ((     True    a(3)
4   100    ())    False    -
5   101    ()(    True    a(4)
6   110    (()    True    a(5)
7   111    (((    True    a(6)
8   1000   ()))   False    -
9   1001   ())(   False    -
10  1010   ()()   True    a(7)
...
20  10100  ()())  False    -
21  10101  ()()(  True    a(13)
...
(End)
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
• Antti Karttunen, Some notes on Catalan's Triangle
• Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A001405, A031443, A036990, A036991, A061855, A014486.

Cf. A237593, A237048, A237591, A237593, A352696.

Programs

Maple

# We use a simple backtracking algorithm: map(op,[seq(NonDivingLatticeSequences(j),j=1..10)]);
NDLS_GLOBAL := []; NonDivingLatticeSequences := proc(n) global NDLS_GLOBAL; NDLS_GLOBAL := []; NonDivingLatticeSequencesAux(0,0,n); RETURN(NDLS_GLOBAL); end;
NonDivingLatticeSequencesAux := proc(x,h,i) global NDLS_GLOBAL; if(0 = i) then NDLS_GLOBAL := [op(NDLS_GLOBAL),x]; else if(h > 0) then NonDivingLatticeSequencesAux((2*x),h-1,i-1); fi; NonDivingLatticeSequencesAux((2*x)+1,h+1,i-1); fi; end;

Mathematica

a061854[n_] := Select[Range[n], !MemberQ[FoldList[#1+If[#2>0, 1, -1]&, 0, IntegerDigits[n, 2]], -1]]
a061854[116] (* Hartmut F. W. Hoft, Mar 29 2022 *)
Select[Range[120],Min[Accumulate[IntegerDigits[#,2]/.(0->-1)]]>=0&] (* Harvey P. Dale, Sep 11 2023 *)