A036990 -id:A036990 - cp's OEIS frontend

A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

0, 2, 10, 12, 42, 44, 50, 52, 56, 170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240, 682, 684, 690, 692, 696, 714, 716, 722, 724, 728, 738, 740, 744, 752, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880, 906, 908, 914

Offset: 0

Wouter Meeussen

The binary Dyck-Language (A063171) in decimal representation.
These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.
Is Sum_{k=1..n} a(k) / n^(5/2) bounded? - Benoit Cloitre, Aug 18 2002
This list is the intersection of A061854 and A031443. - Jason Kimberley, Jan 18 2013
The sequence does start at n = 0, since in the binary interpretation of the Dyck language (e.g., as parenthesizations where "1" stands for "(" and "0" stands for ")") having a(0) = 0 will do since it would stand for the empty string where the "0"s and "1"s are balanced (hence the parentheses are balanced). - Daniel Forgues, Feb 17 2013
It appears that for n>=1 this sequence can be obtained by concatenating the terms of the irregular array whose n-th row length is A000108(n) and that is defined recursively by B(n,0) = A020988(n) and B(n,k) = B(n, k-1) + D(n, k-1) where D(x,y) = (2^(2*(A089309(B(x,y))-1))-1)*(2/3) + 2^A007814(B(x,y)). - Raúl Mario Torres Silva and Michel Marcus, May 01 2020
This encoding is related to the ranking by standard ordered tree numbers in that (1) the binary encoding of trees ordered by standard ranking is given by A358505, while (2) the standard ranking of trees ordered by binary encoding is given by A358523. - Gus Wiseman, Nov 21 2022

			a(19) = 226_10 = 11100010_2 = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:
  0   0
   \ /
    1   0 0  (0)
     \ /   \ /
      1     1
       \   /
         1
Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e., the same bracket expression as above, but surrounded by extra parentheses. This mapping is performed by the Scheme function A014486->parenthesization given below.
From _Gus Wiseman_, Nov 21 2022: (Start)
The terms and corresponding ordered rooted trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
(End)

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 443 (Algorithm P).

Paolo Xausa, Table of n, a(n) for n = 0..23713 (terms 0..2500 from Franklin T. Adams-Watters).
Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 27-34.
Gennady Eremin, Dynamics of balanced parentheses, lexicographic series and Dyck polynomials, arXiv:1909.07675 [math.CO], 2019.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Antti Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
Antti Karttunen, Illustration of 626 initial terms (up to size n=7) with various combinatorial interpretations of Catalan numbers encoded by this sequence.
Antti Karttunen, a089408.c - C program for computing this sequence and many of the related automorphisms.
Antti Karttunen, Some notes on Catalan's Triangle.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
OEIS Wiki, Combinatorial interpretations of Catalan numbers
Frank Ruskey, Algorithmic Solution of Two Combinatorial Problems.
R. P. Stanley, Hipparchus, Plutarch, Schroeder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers.
Paolo Xausa, A Mathematica implementation of Knuth's algorithms P and U.
Index entries for encodings of plane rooted trees (various subsets of this sequence).
Index entries for sequences related to parenthesizing
Index entries for signature-permutations induced by Catalan automorphisms (permutations of natural numbers induced by various bijective operations acting on these structures)
Index entries for the sequences induced by list functions of Lisp (sequences induced by various other operations on these codes or the corresponding structures).

Characteristic function: A080116. Inverse function: A080300.
The terms of binary width 2n are counted by A000108(n). Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
Cf. also A009766, A014137, A071156, A079436, A085184, A213704.
Cf. A020988, A089309, A007814.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
The standard ranking of the ordered tree is A358523.
The depth of the ordered tree is A358550.
Nodes of the ordered tree are counted by A358551.
Cf. A000081, A001263, A358505, A358524.

# Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the a089408.c program for the corresponding C procedures.
CatalanSequences := proc(upto_n) local n,a,r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n,n)/(n+1))-1 do a := [op(a),CatalanUnrank(n,r)]; od; od; return a; end;
CatalanUnrank := proc(n,rr) local r,x,y,lo,m,a; r := (binomial(2*n,n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n,x,y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; return a; end;
Mn := (n,x,y) -> binomial(2*n-x,n-((x+y)/2)) - binomial(2*n-x,n-1-((x+y)/2));
# Alternative:
bin := n -> ListTools:-Reverse(convert(n, base, 2)):
isA014486 := proc(n): local B, s, b; s := 0;
    if n > 0 then
      for b in bin(n) do
          s := s + ifelse(b = 1, 1, -1);
           if 0 > s then return false fi;
      od fi;
  s = 0 end:
select(isA014486, [seq(0..240)]);  # Peter Luschny, Mar 13 2024

cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]
d2b[n_Integer] := IntegerDigits[n, 2]
tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]
Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
(* Alternative code *)
tbQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Count[idn2,1]==Length[idn2]/2&&Min[Accumulate[idn2/.{0->-1}]]>=0]; Join[{0},Select[Range[900],tbQ]] (* Harvey P. Dale, Jul 04 2013 *)
balancedQ[0] = True; balancedQ[n_] := Module[{s = 0}, Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0] ]; A014486 = FromDigits /@ IntegerDigits[Select[Range[0, 1000], balancedQ ]] (* Jean-François Alcover, Mar 05 2016 *)
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; Select[Range[0, 880], A014486Q] (* JungHwan Min, Dec 11 2016 *)
(* Uses Algorithm P from Knuth's TAOCP section 7.2.1.6 - see References and Links. *)
alist[n_] := Block[{a = Flatten[Table[{1, 0}, n]], m = 2*n - 1, j, k},
    FromDigits[#, 2]& /@ Reap[
    While[True,
        Sow[a]; a[[m]] = 0;
        If[a[[m - 1]] == 0,
            a[[--m]] = 1, j = m - 1; k = 2*n - 1;
            While[j > 1 && a[[j]] == 1, a[[j--]] = 0; a[[k]] = 1; k -= 2];
            If[j == 1, Break[]];
            a[[j]] = 1; m = 2*n - 1]
    ]][[2, 1]]];
Join[{{0}, {2}}, Array[alist, 4, 2]] (* Paolo Xausa, Mar 16 2024 *)

isA014486(n)=my(v=binary(n),t=0);for(i=1,#v,t+=if(v[i],1,-1);if(t<0,return(0))); t==0 \\ Charles R Greathouse IV, Jun 10 2011

a_rows(N) = my(a=Vec([[0]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, my(v=if(t, valuation(t, 2), 0)); for(i=0, v, listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 16 2024

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A014486_gen(): # generator of terms
    yield 0
    for l in count(1):
        for s in multiset_permutations('0'*l+'1'*(l-1)):
            c, m = 0, (l<<1)-1
            for i in range(m):
                if s[i] == '1':
                    c += 2
                if cA014486_list = list(islice(A014486_gen(),30)) # Chai Wah Wu, Nov 28 2023

def is_A014486(n) :
    B = bin(n)[2::] if n != 0 else 0
    s = 0
    for b in B :
        s += 1 if b=='1' else -1
        if 0 > s : return False
    return 0 == s
def A014486_list(n): return [k for k in (1..n) if is_A014486(k) ]
A014486_list(888) # Peter Luschny, Aug 10 2012

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004

Added a(0)=0 (which had been removed in June 2011), Joerg Arndt, Feb 27 2013

A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 39, 43, 45, 47, 51, 53, 55, 59, 61, 63, 71, 75, 77, 79, 83, 85, 87, 91, 93, 95, 103, 107, 109, 111, 115, 117, 119, 123, 125, 127, 143, 151, 155, 157, 159, 167, 171, 173, 175, 179, 181, 183, 187, 189, 191, 199, 203

Offset: 1

N. J. A. Sloane

List of binary words that correspond to a valid pairing of parentheses. - Joerg Arndt, Nov 27 2004
This sequence includes as subsequences A000225, A002450, A007583, A036994, A052940, A112627, A113836, A113841, A290114; and also A015521 (without 0), A083713 (without 0), A086224 (without 6), A182512 (without 0). - Gennady Eremin, Nov 27 2021 and Aug 26 2023
Partial differences are powers of 2 (cf. A367626, A367627). - Gennady Eremin, Dec 23 2021
This is the sequence A030101(A014486(n)), n >= 0, sorted into ascending order. See A014486 for more references, illustrations, etc., concerning Dyck paths and other associated structures enumerated by the Catalan numbers. - Antti Karttunen, Sep 25 2023
The terms in this sequence with a given length in base 2 are counted by A001405. For example, the number of terms of bit length k=5 (these are 19, 21, 23, 27, 29, and 31) is equal to A001405(k-1) = A001405(4) = 6. - Gennady Eremin, Nov 07 2023

			From _Joerg Arndt_, Dec 05 2021: (Start)
List of binary words with parentheses for those in the sequence (indicated by P). The binary words are scanned starting from the least significant bit, while the parentheses words are written left to right:
     Binary   Parentheses (if the value is in the sequence)
00:  ..... P  [empty string]
01:  ....1 P   ()
02:  ...1.
03:  ...11 P   (())
04:  ..1..
05:  ..1.1 P   ()()
06:  ..11.
07:  ..111 P   ((()))
08:  .1...
09:  .1..1
10:  .1.1.
11:  .1.11 P   (()())
12:  .11..
13:  .11.1 P   ()(())
14:  .111.
15:  .1111 P   (((())))
16:  1....
17:  1...1
18:  1..1.
19:  1..11 P   (())()
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..13496 (all terms < 2^16; first 1000 terms from Reinhard Zumkeller)
Joerg Arndt, Matters Computational (The Fxtbook), section 1.28, pp. 78-80.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318, 2023. See pp. 1-2, 4, 7-11, 13-14.
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp 2-5, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
Index entries for sequences related to binary expansion of n

Cf. A350577 (primes subsequence).
See also A014486, A030101, A036988, A036990, A036992. A036994 is a subset (requires the count of zeros to be strictly less than the count of 1's).
See also A030308, A000225, A002450, A007583, A350346, A367625, A367626 & A367627 (first differences).

a036991 n = a036991_list !! (n-1)
a036991_list = filter ((p 1) . a030308_row) [0..] where
   p     []    = True
   p ones (0:bs) = ones > 1 && p (ones - 1) bs
   p ones (1:bs) = p (ones + 1) bs
-- Reinhard Zumkeller, Jul 31 2013

q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
      for i to nops(l) do t:= t-1+2*l[i];
        if t<0 then return false fi
      od: true
    end:
select(q, [$0..300])[];  # Alois P. Heinz, Oct 09 2019

moreOnesRLQ[n_Integer] := Module[{digits, len, flag = True, iter = 1, ones = 0, zeros = 0}, digits = Reverse[IntegerDigits[n, 2]]; len = Length[digits]; While[flag && iter < len, If[digits[[iter]] == 1, ones++, zeros++]; flag = ones >= zeros; iter++]; flag]; Select[Range[0, 203], moreOnesRLQ] (* Alonso del Arte, Sep 21 2011 *)
Join[{0},Select[Range[210],Min[Accumulate[Reverse[IntegerDigits[#,2]]/.{0->-1}]]>-1&]] (* Harvey P. Dale, Apr 18 2014 *)

select( {is_A036991(n,c=1)=!n||!until(!n>>=1,(c-=(-1)^bittest(n,0))||return)}, [0..99]) \\ M. F. Hasler, Nov 26 2021

def ok(n):
    if n == 0: return True # by definition
    count = {"0": 0, "1": 0}
    for bit in bin(n)[:1:-1]:
        count[bit] += 1
        if count["0"] > count["1"]: return False
    return True
print([k for k in range(204) if ok(k)]) # Michael S. Branicky, Nov 25 2021

from itertools import count, islice
def A036991_gen(): # generator of terms
    yield 0
    for n in count(1):
        s = bin(n)[2:]
        c, l = 0, len(s)
        for i in range(l):
            c += int(s[l-i-1])
            if 2*c <= i:
                break
        else:
            yield n
A036991_list = list(islice(A036991_gen(),20)) # Chai Wah Wu, Dec 30 2021

If a(n) = A000225(k) for some k, then a(n+1) = a(n) + A060546(k). - Gennady Eremin, Nov 07 2023

More terms from Erich Friedman
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
Offset corrected and example adjusted accordingly by Reinhard Zumkeller, Jul 31 2013

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.

Original entry on oeis.org

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

Antti Karttunen, May 11 2001

"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

			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)

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

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

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

# 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;

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 *)

A036988 Has simplest possible tree complexity of all transcendental sequences.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.

Cf. A036989. Characteristic function of A036990.

a036988 = a063524 . a036989  -- Reinhard Zumkeller, Jul 31 2013

(* b = A036989 *) b[0] = 1; b[n_?EvenQ] := b[n] = Max[b[n/2] - 1, 1]; b[n_] := b[n] = b[(n-1)/2] + 1; a[n_] := Boole[b[n] == 1]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 05 2013, after Reinhard Zumkeller *)

a(n) = 1 iff, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.

a(n) = A063524(A036989(n)). - Reinhard Zumkeller, Jul 31 2013

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A036989 Read binary expansion of n from the right; keep track of the excess of 1's over 0's that have been seen so far; sequence gives 1 + maximum(excess of 1's over 0's).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 3, 3, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 6, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 3, 3, 5, 1, 2, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 7, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 3, 3, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 6, 1, 2, 1, 3, 1, 2, 2, 4, 1

Offset: 0

N. J. A. Sloane

Associated with A036988 (Remark 4 of the reference).

			59 in binary is 111011, excess from right to left is 1,2,1,2,3,4, maximum is 4, so a(59) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.
Index entries for sequences related to binary expansion of n

Cf. A036988, A036990, A126387.

Cf. A010060.

import Data.List (transpose)
a036989 n = a036989_list !! n
a036989_list = 1 : concat (transpose
   [map (+ 1) a036989_list, map ((max 1) . pred) $ tail a036989_list])
-- Reinhard Zumkeller, Jul 31 2013

a[0] = 1; a[n_?EvenQ] := a[n] = Max[a[n/2] - 1, 1]; a[n_] := a[n] = a[(n-1)/2] + 1; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 05 2013, after Franklin T. Adams-Watters *)

a(n) = 1 iff, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's: a(A036990(n)) = 1.
a(0) = 1, a(2n) = max(a(n) - 1, 1), a(2n+1) = a(n) + 1. - Franklin T. Adams-Watters, Dec 26 2006
Equals inverse Moebius transform (A051731) of A010060, the Thue-Morse sequence starting with "1": (1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, ...). - Gary W. Adamson, May 13 2007

Edited by Joshua Zucker, May 11 2006

A036993 Numbers n with property that reading from right to left in the binary expansion of n, the number of 0's always stays ahead of the number of 1's.

Original entry on oeis.org

0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 128, 132, 136, 144, 148, 152, 160, 164, 168, 176, 192, 196, 200, 208, 224, 256, 260, 264, 272, 276, 280, 288, 292, 296, 304, 320, 324, 328, 336, 340, 344, 352, 356, 360, 368, 384

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Cf. A036990, A036991, A036992, A036994.

Cf. A030308.

a036993 n = a036993_list !! (n-1)
a036993_list = filter ((p 0) . a030308_row) [0..] where
   p _     []     = True
   p zeros (0:bs) = p (zeros + 1) bs
   p zeros (1:bs) = zeros > 1 && p (zeros - 1) bs
-- Reinhard Zumkeller, Jul 31 2013

Select[Range[0,400],Min[Accumulate[Reverse[IntegerDigits[#,2]]/.{0->1, 1->-1}]]>0&] (* Harvey P. Dale, Aug 25 2013 *)

More terms from Erich Friedman

A036994 Numbers k with the property that reading from right to left in the binary expansion of k, the number of 1's always stays ahead of the number of 0's.

Original entry on oeis.org

1, 3, 7, 11, 15, 23, 27, 31, 39, 43, 47, 55, 59, 63, 79, 87, 91, 95, 103, 107, 111, 119, 123, 127, 143, 151, 155, 159, 167, 171, 175, 183, 187, 191, 207, 215, 219, 223, 231, 235, 239, 247, 251, 255, 287, 303, 311, 315, 319, 335, 343, 347, 351, 359, 363, 367

Offset: 1

N. J. A. Sloane

Even numbers can't appear in this sequence. - Alonso del Arte, Sep 21 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A036990, A036991, A036992, A036993.

Cf. A030308.

a036994 n = a036994_list !! (n-1)
a036994_list = filter ((p 0) . a030308_row) [1, 3 ..] where
   p ones []    = ones > 0
   p ones (0:bs) = ones > 1 && p (ones - 1) bs
   p ones (1:bs) = p (ones + 1) bs
-- Reinhard Zumkeller, Aug 01 2013

aheadOnesRLQ[n_Integer] := Module[{digits, len, flag = True, iter = 1, ones = 0, zeros = 0}, digits = Reverse[IntegerDigits[n, 2]]; len = Length[digits]; While[flag && iter < len, If[digits[[iter]] == 1, ones++, zeros++]; flag = ones > zeros; iter++]; flag]; Select[Range[1, 201, 2], aheadOnesRLQ] (* Alonso del Arte, Sep 21 2011 *)
Select[Range[400],Min[Accumulate[Reverse[IntegerDigits[#,2]/. (0->-1)]]]> 0&] (* Harvey P. Dale, Apr 23 2016 *)

ok(x)={if(x<1,return(0));my(c=logint(x,2),c0=0,c1=0);for(i=0,c,if(bittest(x,i),c1++,c0++);if(c1<=c0,return(0)));1} \\ for(n=1,367,if(ok(n),print1(n,", "))) - Ruud H.G. van Tol, Sep 14 2022

from itertools import count, islice
def A036994_gen(startvalue=0): # generator of terms >= startvalue
    for n in count(max(startvalue,0)):
        s = bin(n)[2:]
        c, l = 0, len(s)
        for i in range(l):
            c += int(s[l-i-1])
            if 2*c <= i + 1:
                break
        else:
            yield n
A036994_list = list(islice(A036994_gen(),20)) # Chai Wah Wu, Dec 31 2021

More terms from Erich Friedman

cp's OEIS Frontend

A036992 Numbers not in A036990 nor A036991.

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A125086 Even numbers not in A036990.

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A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

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A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

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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.

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A036988 Has simplest possible tree complexity of all transcendental sequences.

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A036989 Read binary expansion of n from the right; keep track of the excess of 1's over 0's that have been seen so far; sequence gives 1 + maximum(excess of 1's over 0's).

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