A003754 -id:A003754 - cp's OEIS frontend

A052499 If n is in the sequence then so are 2n and 4n-1.

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 15, 16, 22, 23, 24, 27, 28, 30, 31, 32, 43, 44, 46, 47, 48, 54, 55, 56, 59, 60, 62, 63, 64, 86, 87, 88, 91, 92, 94, 95, 96, 107, 108, 110, 111, 112, 118, 119, 120, 123, 124, 126, 127, 128, 171, 172, 174, 175, 176, 182, 183, 184, 187

Offset: 0

Henry Bottomley, Mar 15 2000

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = 1 + A003754.

			a(9)=14 is in the sequence because 14=2*(4*(2*1)-1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
T. Karki, A. Lacroix, M. Rigo, On the recognizability of self-generating sets, JIS 13 (2010) #10.2.2.
C. Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.

Cf. A001911, A003754.

import Data.Set (singleton, deleteFindMin, insert)
a052499 n = a052499_list !! n
a052499_list = f $ singleton 1 where
   f s = m : f (insert (2*m) $ insert (4*m-1) s') where
      (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jul 06 2011

1 + Select[ Range[0, 200], FreeQ[ IntegerDigits[#, 2], {_, 0, 0, _} ] & ] (* Jean-François Alcover, Jan 20 2012, after J.-P. Allouche *)
a[1] = 1; a[n_] := a[n] = a[n - 1] + Ceiling[2^IntegerExponent[a[n - 1], 2]/3]; Array[a, 200] (* Birkas Gyorgy, May 30 2012 *)

from itertools import count, islice
def A052499_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not '00' in bin(n-1),count(max(startvalue,1)))
A052499_list = list(islice(A052499_gen(),20)) # Chai Wah Wu, Feb 12 2025

a(A001911(n)) = 2^n.

A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111

Offset: 0

Reinhard Zumkeller, May 28 2005

Union of A003754 and A003714, complement of A107911;
a(A023548(n+2)) = A052940(n+1) for n>0;
a(A001924(n)) = A000225(n) = 2^n - 1;
a(A000126(n)) = A000079(n) = 2^n for n>0;
A107910(n) = a(n+1) - a(n).

Ivan Neretin, Table of n, a(n) for n = 0..10000
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, 2014.
Index entries for 2-automatic sequences.

Cf. A007088, A107907.

foreach $n(1..100){$_=sprintf("%b",$n); print "$n\n" if !m/11/||!m/00/}
# Ivan Neretin, May 01 2016

A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 13, 10, 21, 8, 45, 26, 93, 20, 109, 42, 85, 16, 173, 90, 477, 52, 957, 186, 733, 40, 749, 218, 1501, 84, 877, 170, 341, 32, 685, 346, 3549, 180, 12221, 954, 7133, 104, 14269, 1914, 49021, 372, 28605, 1466, 5853, 80, 5869, 1498, 30685, 436, 61373, 3002, 23517, 168, 12013, 1754, 24029, 340, 7021, 682, 1365

Offset: 0

Antti Karttunen, Oct 07 2016

Sequence encodes Stern polynomials (see A125184, A260443) with "unary-binary method" where any nonzero coefficient c > 0 is encoded as a run of c 1-bits, separated from neighboring 1-runs by exactly one zero (this follows because A260442 is a subsequence of A073491). See the examples.

Terms which are not multiples of 4 form a subset of A003754, or in other words, each term is 2^k * {a term from a certain subsequence of A247648}, which subsequence remains to be determined.

			n    Stern polynomial       Encoded as              a(n)
                            "unary-binary" number   (-> decimal)
----------------------------------------------------------------
0    B_0(x) = 0                     "0"               0
1    B_1(x) = 1                     "1"               1
2    B_2(x) = x                    "10"               2
3    B_3(x) = x + 1               "101"               5
4    B_4(x) = x^2                 "100"               4
5    B_5(x) = 2x + 1             "1101"              13
6    B_6(x) = x^2 + x            "1010"              10
7    B_7(x) = x^2 + x + 1       "10101"              21
8    B_8(x) = x^3                "1000"               8
9    B_9(x) = x^2 + 2x + 1     "101101"              45

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A087808 (a left inverse), A156552, A260443, A277189 (odd bisection).

Cf. also A000079, A000120, A000225, A002450, A002487, A003754, A073491, A247648, A260442, A277010, A277012, A276081.

(define (A277020 n) (A156552 (A260443 n)))
;; Another implementation, more practical to run:
(define (A277020 n) (list_of_numbers_to_unary_binary_representation (A260443as_index_lists n)))
(definec (A260443as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_index_lists (/ n 2)))) (else (add_two_lists (A260443as_index_lists (/ (- n 1) 2)) (A260443as_index_lists (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(define (list_of_numbers_to_unary_binary_representation nums) (let loop ((s 0) (nums nums) (b 1)) (cond ((null? nums) s) (else (loop (+ s (* (A000225 (car nums)) b)) (cdr nums) (* (A000079 (+ 1 (car nums))) b))))))

a(n) = A156552(A260443(n)).
Other identities. For all n >= 0:
A087808(a(n)) = n.
A000120(a(n)) = A002487(n).
a(2n) = 2*a(n).
a(2^n) = 2^n.
a(A000225(n)) = A002450(n).

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

A004743 Numbers whose binary expansion does not contain 110.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 47, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 79, 80, 81, 82, 83, 84, 85, 87, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139

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), A004742 (no 101), A003726 (no 111).

a004743 n = a004743_list !! (n-1)
a004743_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 6 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 140], !StringContainsQ[IntegerString[#, 2], "110"] &] (* Amiram Eldar, Feb 13 2022 *)
Select[Range[0,150],SequenceCount[IntegerDigits[#,2],{1,1,0}]==0&] (* Harvey P. Dale, Mar 14 2025 *)

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>5, if(bitand(n,7)==6, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 5.126608057149204485684180689064467269298250594297584060475240185531109866051... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 1, 7, 0, 1, 2, 3, 12, 0, 1, 2, 0, 4, 5, 6, 20, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 33, 0, 1, 2, 0, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 13, 17, 18, 19, 54, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 12, 0, 1, 2, 0, 4, 5, 6, 20, 21, 22, 23, 24

Offset: 0

Rémy Sigrist, Aug 27 2022

The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):
- in the Zeckendorf representation (or greedy Fibonacci representation):
  - Fibonacci numbers are as big as possible (see A035517),
  - and the corresponding binary encoding, A003714(n),
    cannot have two consecutive 1's;
- in the dual Zeckendorf representation (or lazy Fibonacci representation):
  - Fibonacci numbers are as small as possible (see A112309),
  - and the corresponding binary encoding, A003754(n+1),
    cannot have two consecutive nonleading 0's.
See A356326 for a similar sequence.

			For n = 28:
- using F(k) = A000045(k),
- the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
- the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
- F(5) and F(3) appear in both representations,
- so a(28) = F(5) + F(3) = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.

PARI
```
\\ See Links section.
```

a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).
a(n) = 0 iff n belongs to A331467.
a(n) = n iff n belongs to A000071.

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

A357137 Maximal run-length of the n-th composition in standard order; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 3, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 3, 2

Offset: 0

Gus Wiseman, Sep 18 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, 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.

			Composition 92 in standard order is (2,1,1,3), so a(92) = 2.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for more sequences related to standard compositions.
The version for Heinz numbers of partitions is A051903, for parts A061395.
For parts instead of run-lengths we have A333766, minimal A333768.
The opposite (minimal) version is A357138.
For first instead of maximal we have A357180, last A357181.
Cf. A000120, A001511, A003754, A029931, A051904, A055396, A056239, A070939, A286470, A356844, A357136.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Max[Length/@Split[stc[n]]]],{n,0,100}]

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

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A052499 If n is in the sequence then so are 2n and 4n-1.

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A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

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A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).

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A004742 Numbers whose binary expansion does not contain 101.

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A004743 Numbers whose binary expansion does not contain 110.

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A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

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A004744 Numbers whose binary expansion does not contain 011.

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