A004743 -id:A004743 - cp's OEIS frontend

A114994 Numbers whose binary representation has monotonically decreasing sizes of groups of zeros (including zero-length groups between adjacent ones).

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 79, 85, 87, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 143, 146, 147, 149, 151, 159, 170, 171, 175

Offset: 0

Franklin T. Adams-Watters, Feb 22 2006

Numbers whose binary representation avoids the sequences 110, 10100, 1001000, etc. Represents partitions. Start with empty partition and process each bit from left to right: if a zero, increase the size of the smallest part; if one, add a new size 1 part. This generates the partitions in Mathematica order. Can be regarded as a table with row lengths A000041(n); values 2^n <= a(m) < 2^(n+1) are in row n, representing the partitions of n. (Interpreting arbitrary binary numbers in this way generates compositions [also known as ordered partitions]; these are the compositions where the part sizes are in decreasing order of size.)
From Vladimir Shevelev, Dec 09 2013: (Start)
Every number in binary is a concatenation of parts of the form 10...0 with k>=0 zeros. For example, 5=(10)(1), 11=(10)(1)(1), 7=(1)(1)(1). Define c-multiplication [*] by adding multiplicities of parts (ordering by nonincreasing numbers of 0's). For example, 5[*]3=(10)(1)(1)(1)=23. Two numbers we call equivalent if they have the same parts with the same multiplicities. So 6~5, 12~9, 14~13~11.
The sequence lists equivalence classes of integers, choosing the minimal representative in each.
Note that, for two terms x,y we have x[*]y=y[*]x (commutativity), and for three terms x,y,z we have x[*](y[*]z)= (x[*]y)[*]z (associativity). 0 is the unit, i.e., 0[*]x=x. Moreover, one can consider different parts, i.e., {2^n} as "c-primes". Then every term is a unique "c-product" of "c-powers" of c-primes. For example, 7=(1)^3, 10=(10)^2, etc.
Further, one can naturally introduce "c-notions": c-divisor, c-divisibility, greatest common c-divisor of several numbers and least common c-multiple, Euler c-totient function (with notion of "r is c-prime to m"), etc.
Let x[+]y denote usual sum x+y in which we order parts over nonincreasing number of zeros. Then, of course, A114994 is closed over such operation. Then a(n+1) = a(n)[+]k, where k is the least number such that a(n)[+]k > a(n). For example, since a(10)=11, we have 11[+]1=9, 11[+]2=11, 11[+]3=11, 11[+]4=15>11. So, a(11)=15.
(End)

			21 is included, binary 10101 has group sizes 1,1,0; 22 is not, binary 10110 has group sizes 1,0,1, which includes an increase.
Applying bits of 21 in order gives sequence of partitions: [], [1], [2], [2,1], [2^2], [2^2,1], so 21 represents the partition [2^2,1].
From _Omar E. Pol_, Aug 04 2013: (Start)
The positive terms written as an irregular triangle begins:
   1;
   2,  3;
   4,  5,  7;
   8,  9, 10, 11, 15;
  16, 17, 18, 19, 21, 23, 31;
  32, 33, 34, 35, 36, 37, 39, 42, 43, 47, 63;
  64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 79, 85, 87, 95, 127;
  ...
Column 1 is A000079. Right border gives A000225, n >= 1.
T(n,k) represents the k-th partition of n. Example: for n = 5 the seven partitions of 5 (in Mathematica order) are represented in three ways as shown below. The last column (16, 17, 18, 19, 21, 23, 31) is also the 5th row of triangle.
-----------------------------------
Partitions      Binary     Decimal
of 5            number      value
-----------------------------------
5               10000        16
4+1             10001        17
3+2             10010        18
3+1+1           10011        19
2+2+1           10101        21
2+1+1+1         10111        23
1+1+1+1+1       11111        31
(End)
From _Peter J. C. Moses_, Dec 09 2013: (Start)
Let us illustrate an algorithm of calculation of all terms in interval of the form [2^k,2^(k+1)). Let k=5. Consider all integer partitions of 5+1=6 ordered over decreasing of maximal parts (see algorithm IntegerPartitions). We have: {{6},{5,1},{4,2},{4,1,1},{3,3},{3,2,1},{3,1,1,1},{2,2,2},{2,2,1,1},{2,1,1,1,1},{1,1,1,1,1,1}}.
Now for every number, i, replace it with 1 followed by (i-1) 0's. So that becomes: {{1,0,0,0,0,0},{1,0,0,0,0,1},{1,0,0,0,1,0},{1,0,0,0,1,1},{1,0,0,1,0,0},{1,0,0,1,0,1},{1,0,0,1,1,1},{1,0,1,0,1,0},{1,0,1,0,1,1},{1,0,1,1,1,1},{1,1,1,1,1,1}}.
Finally, reading these as binary numbers with transformation of them into decimal, we obtain all terms in interval [32,64): {32,33,34,35,36,37,39,42,43,47,63}.
(End)

Peter J. C. Moses, Table of n, a(n) for n = 0..4999

Cf. A004743, A080577, A000041, A232897, A233027, A000070.

Cf. also A227739, A227183 and permutation pair A229119/A229120 for another system of encoding unordered partitions in the binary representation of n.

Select[Range[0, 200], FromDigits[Flatten[Sort[Split[IntegerDigits[#, 2], #1>#2||#2==0&], Length[#1]>Length[#2]&]], 2]==#&] (* Peter J. C. Moses, Dec 04 2013 *)
f:=Map[IntegerDigits[2^(#-1), 2]&, #]&; Flatten[Map[Map[FromDigits[#, 2]&, Map[Flatten, f[IntegerPartitions[#]]]]&, Range[0, 10]]] (* Peter J. C. Moses, Dec 05 2013 *)

is(n, k=0)=if(n==0, return(1)); my(e=valuation(n, 2)); if(e>(e+1), e)) \\ Charles R Greathouse IV, Dec 05 2013

For n>=0, 2n+1 is in the sequence iff n is in the sequence. For n>0, 2n is in the sequence iff both n is the sequence and, for some k>=0, n is congruent to 2^k mod 4^(k+1).

Number terms in interval [2^(n-1), 2^n) is A000041(n); number terms <2^n is A000070(n). - Vladimir Shevelev, Dec 06 2013

A003754 Numbers with no adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A003726 Numbers with no 3 adjacent 1's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 80, 81, 82

Offset: 1

N. J. A. Sloane

Positions of zeros in A014082. Could be called "tribbinary numbers" by analogy with A003714. - John Keith, Mar 07 2022

The sequence of Tribbinary numbers can be constructed by writing out the Tribonacci representations of nonnegative integers and then evaluating the result in binary. These numbers are similar to Fibbinary numbers A003714, Fibternary numbers A003726, and Tribternary numbers A356823. The number of Tribbinary numbers less than any power of two is a Tribonacci number. We can generate Tribbinary numbers recursively: Start by adding 0 and 1 to the sequence. Then, if x is a number in the sequence add 2x, 4x+1, and 8x+3 to the sequence. The n-th Tribbinary number is even if the n-th term of the Tribonacci word is a. Respectively, the n-th Tribbinary number is of the form 4x+1 if the n-th term of the Tribonacci word is b, and the n-th Tribbinary number is of the form 8x+3 if the n-th term of the Tribonacci word is c. Every nonnegative integer can be written as the sum of two Tribbinary numbers. Every number has a Tribbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A278038 (binary), A063037, A000073, A014082 (number of 111).
Cf. A004781 (complement).
Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110).

a003726 n = a003726_list !! (n - 1)
a003726_list = filter f [0..] where
   f x = x < 7 || (x `mod` 8) < 7 && f (x `div` 2)
-- Reinhard Zumkeller, Jun 03 2012

Select[Range[0, 82], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* Michael De Vlieger, Dec 23 2019 *)

is(n)=!bitand(bitand(n, n<<1), n<<2) \\ Charles R Greathouse IV, Feb 11 2017

There are A000073(n+3) terms of this sequence with at most n bits. In particular, a(A000073(n+3)+1) = 2^n. - Charles R Greathouse IV, Oct 22 2021

Sum_{n>=2} 1/a(n) = 9.516857810319139410424631558212354346868048230248717360943194590798113163384... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 33, 34, 36, 64, 65, 66, 68, 72, 73, 128, 129, 130, 132, 136, 137, 144, 145, 146, 256, 257, 258, 260, 264, 265, 272, 273, 274, 288, 289, 290, 292, 512, 513, 514, 516, 520, 521, 528, 529, 530, 544, 545, 546, 548, 576, 577, 578, 580

Offset: 0

Antti Karttunen, Mar 30 1999

No more than one 1-bit in each bit triple.
All terms satisfy A048727(n) = 7*n.
Constructed from A000930 in the same way as A003714 is constructed from A000045.
It appears that n is in the sequence if and only if C(7n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
The conjecture by Benoit is correct. This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p. - Franklin T. Adams-Watters, Oct 06 2009
Appears to be the set of numbers x such that (x AND 5*x) = x and (x OR 3*x)/x = 3. - Gary Detlefs, Jun 08 2024

G. C. Greubel, Table of n, a(n) for n = 0..1275
Sebastian Karlsson, Walnut code that verifies the conjectures of Paul D. Hanna
Walnut can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR
Index entries for 2-automatic sequences.

Subsequence of A048716.

Cf. A003726, A004742, A004743, A004744, A048717, A048718, A048719, A048730, A048733, A115422, A115423, A115424.

Reap[Do[If[OddQ[Binomial[7n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {_, 1, 0, 1, _}|{_, 1, 1, _}]];
Select[Range[0, 580], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n, 6*n) \\ Charles R Greathouse IV, Oct 03 2016

for my $k (0..580) { print "$k, " if sprintf("%b", $k) =~ m{^(100(0)*)*(0|1|10)?$}; } # Georg Fischer, Jun 26 2021

import re
def ok(n): return re.fullmatch('(100(0)*)*(0|1|10)?', bin(n)[2:]) != None
print(list(filter(ok, range(581)))) # Michael S. Branicky, Jun 26 2021

a(0) = 0, a(n) = (2^(invfoo(n)-1))+a(n-foo(invfoo(n))), where foo(n) is foo(n-1) + foo(n-3) (A000930) and invfoo is its "integral" (floored down) inverse.
a(n) XOR 6*a(n) = 7*a(n); 3*a(n) XOR 4*a(n) = 7*a(n); 3*a(n) XOR 5*a(n) = 6*a(n); (conjectures). - Paul D. Hanna, Jan 22 2006
The conjectures can be verified using the Walnut theorem-prover (see links). - Sebastian Karlsson, Dec 31 2022

Definition corrected by Georg Fischer, Jun 26 2021

A003796 Numbers with no 3 adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.

Complement of A004779.
Cf. A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Cf. A063037, A007088.

a003796 n = a003796_list !! (n-1)
a003796_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 0 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,100],SequenceCount[IntegerDigits[#,2],{0,0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2015 *)

is(n)=while(n>7,if(bitand(n,7)==0,return(0));n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 9.829256652701616366441622119246549956902006567009112470631751387637507184399... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004742 Numbers whose binary expansion does not contain 101.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 19, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 48, 49, 50, 51, 56, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 76, 78, 79, 96, 97, 98, 99, 100, 102, 103, 112, 113, 114, 115, 120, 121, 124, 126, 127

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004743 (no 110), A003726 (no 111).

a004742 n = a004742_list !! (n-1)
a004742_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 5 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 130], !StringContainsQ[IntegerString[#, 2], "101"] &] (* Amiram Eldar, Feb 13 2022 *)

is(n)=n=binary(n);for(i=3,#n,if(n[i]&&n[i-2]&&!n[i-1],return(0))); 1 \\ Charles R Greathouse IV, Mar 26 2013

is(n)=while(n>4, if(bitand(n,7)==5, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

is(n)=!bitand(bitand(n,n>>2),bitneg(n>>1)) \\ Charles R Greathouse IV, Oct 28 2021

searchLE(S,x)=my(t=setsearch(S,x)); if(t,t,setsearch(S,x,1)-1); \\ finds last element <= x
expand(~v, lim)=my(b=exponent(v[#v]+1), B=1<lim, listpop(~v));
list(lim)=lim\=1; if(lim<5, return(if(lim<0,[],[0..lim]))); my(v=List([0..3])); for(b=3,exponent(lim+1), expand(~v, 2^b-1)); expand(~v, lim); Vec(v)

Sum_{n>=2} 1/a(n) = 6.198475910942069028389983717965787117743378665090593775808705963863146498248... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004744 Numbers whose binary expansion does not contain 011.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 48, 49, 50, 52, 53, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 96, 97, 98, 100, 101, 104, 105

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004744 n = a004744_list !! (n-1)
a004744_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 3 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,110],!MemberQ[Partition[IntegerDigits[#,2],3,1],{0,1,1}]&] (* Harvey P. Dale, Oct 15 2013 *)

is(n)=n=binary(n);for(i=3,#n,if(n[i]&&n[i-1]&&!n[i-2], return(0)));1 \\ Charles R Greathouse IV, Mar 26 2013

is(n)=while(n>10, if(bitand(n,7)==3, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 6.084750966700965350831194838591995529232464122788387705746226526437263331240... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004745 Numbers whose binary expansion does not contain 001.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 40, 42, 43, 44, 45, 46, 47, 48, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 80, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 104, 106, 107, 108, 109, 110

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004745 n = a004745_list !! (n-1)
a004745_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 1 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 110], ! StringContainsQ[IntegerString[#, 2], "001"] &] (* Amiram Eldar, Feb 13 2022 *)
Select[Range[0,120],SequenceCount[IntegerDigits[#,2],{0,0,1}]==0&] (* Harvey P. Dale, Jul 05 2024 *)

is(n)=n=binary(n);for(i=4,#n,if(n[i]&&!n[i-1]&&!n[i-2], return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

is(n)=while(n>8, if(bitand(n,7)==1, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 5.808784664093998434778841785199192904637860758506854276321167162567685504669... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004746 Numbers whose binary expansion does not contain 010.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 38, 39, 44, 45, 46, 47, 48, 49, 51, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 70, 71, 76, 77, 78, 79, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 102

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004746 n = a004746_list !! (n-1)
a004746_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 2 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,110],SequenceCount[IntegerDigits[#,2],{0,1,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Oct 19 2015 *)

is(n)=n=binary(n);for(i=4,#n,if(!n[i]&&n[i-1]&&!n[i-2], return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

is(n)=while(n>9, if(bitand(n,7)==2, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 7.338340181978485860731253930056466995425939377143636935044890325770833657631... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

cp's OEIS Frontend

A114994 Numbers whose binary representation has monotonically decreasing sizes of groups of zeros (including zero-length groups between adjacent ones).

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A003754 Numbers with no adjacent 0's in binary expansion.

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A003726 Numbers with no 3 adjacent 1's in binary expansion.

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A048715 Binary expansion matches (100(0)*)*(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

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A003796 Numbers with no 3 adjacent 0's in binary expansion.

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A004742 Numbers whose binary expansion does not contain 101.

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A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).