A063037 -id:A063037 - cp's OEIS frontend

A357522 Reverse run lengths in binary expansions of terms of A063037: for n >= 0, a(n) is the unique k such that A063037(1+k) = A056539(A063037(1+n)).

0, 1, 2, 3, 6, 5, 4, 7, 8, 11, 10, 9, 16, 17, 18, 15, 12, 13, 14, 19, 32, 23, 22, 21, 24, 31, 28, 27, 26, 29, 30, 25, 20, 33, 42, 49, 48, 43, 44, 47, 50, 41, 34, 37, 38, 53, 52, 39, 36, 35, 40, 51, 46, 45, 74, 75, 84, 65, 58, 59, 64, 85, 86, 63, 60, 57, 66, 83

Offset: 0

Rémy Sigrist, Oct 02 2022

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 42:
- A063037(1+42) = 86,
- the binary expansion of 86 is "1010110",
- reversing run lengths yields "1001010",
- this corresponds to 74 = A063037(1+34),
- hence a(42) = 34.

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A357523 for a similar sequence.

Cf. A044918, A056539, A063037.

PARI
```
See Links section.
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a(n) = n iff n = 0 or A044813(1+n) belongs to A044918.

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

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

A179888 Starting with a(1)=2: if m is a term then also 4m+1 and 4m+2.

Original entry on oeis.org

2, 9, 10, 37, 38, 41, 42, 149, 150, 153, 154, 165, 166, 169, 170, 597, 598, 601, 602, 613, 614, 617, 618, 661, 662, 665, 666, 677, 678, 681, 682, 2389, 2390, 2393, 2394, 2405, 2406, 2409, 2410, 2453, 2454, 2457, 2458, 2469, 2470, 2473, 2474, 2645, 2646, 2649

Offset: 1

Reinhard Zumkeller, Jul 31 2010

nonn

0 -> 01 and 1 -> 10 in binary representation of n;
intersection of A032925 and A053754;
subsequence of A063037;
A000120(a(n))=A023416(a(n))=A070939(n); A070939(a(n))=2*A070939(n).

			__ n | __ bin(n) || ___ bin(a(n)) | base-4(a(n)) | __ a(n)
-----|-----------||---------------|--------------|---------
.. 1 | ....... 1 || .......... 10 | .......... 2 | ..... 2;
.. 2 | ...... 10 || ........ 1001 | ......... 21 | ..... 9;
.. 3 | ...... 11 || ........ 1010 | ......... 22 | .... 10;
.. 4 | ..... 100 || ...... 100101 | ........ 211 | .... 37;
.. 5 | ..... 101 || ...... 100110 | ........ 212 | .... 38;
.. 6 | ..... 110 || ...... 101001 | ........ 221 | .... 41;
.. 7 | ..... 111 || ...... 101010 | ........ 222 | .... 42;
.. 8 | .... 1000 || .... 10010101 | ....... 2111 | ... 149;
.. 9 | .... 1001 || .... 10010110 | ....... 2112 | ... 150;
. 10 | .... 1010 || .... 10011001 | ....... 2121 | ... 153;
. 11 | .... 1011 || .... 10011010 | ....... 2122 | ... 154;
. 12 | .... 1100 || .... 10100101 | ....... 2211 | ... 165;
. 13 | .... 1101 || .... 10100110 | ....... 2212 | ... 166;
. 14 | .... 1110 || .... 10101001 | ....... 2221 | ... 169;
. 15 | .... 1111 || .... 10101010 | ....... 2222 | ... 170;
. 16 | ... 10000 || .. 1001010101 | ...... 21111 | ... 597;
. 17 | ... 10001 || .. 1001010110 | ...... 21112 | ... 598;
. 18 | ... 10010 || .. 1001011001 | ...... 21121 | ... 601;
. 19 | ... 10011 || .. 1001011010 | ...... 21122 | ... 602;
. 20 | ... 10100 || .. 1001100101 | ...... 21211 | ... 613.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quaternary
Index entries for sequences related to binary expansion of n

Cf. A132679, A196168.

a179888 n = a179888_list !! (n-1)
a179888_list = 2 : f a179888_list where
  f (x:xs) = x' : x'' : f (xs ++ [x',x'']) where x' = 4*x+1; x'' = x' + 1
-- Reinhard Zumkeller, Oct 29 2011

a:= n-> 1+(n mod 2)+`if`(n<2, 0, 4*a(iquo(n, 2))):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 15 2024

Union@ Flatten@ NestList[ {4 # + 1, 4 # + 2} &, 2, 5] (* Robert G. Wilson v, Aug 16 2011 *)

def A179888(n): return ((1<<(n.bit_length()<<1))-1)//3+int(bin(n)[2:],4) # Chai Wah Wu, Jul 16 2024

a(n) = 4*a(floor(n/2)) + n mod 2 + 1 for n>1;

a(n) = SUM((bit(k)+1)*4^k: 0<=k<=L), where bit() and L such that n=SUM(bit(k)*2^k: 0<=k<=L).

A286262 Numbers whose binary expansion is a cubefree string.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 27, 36, 37, 38, 41, 43, 44, 45, 50, 51, 52, 53, 54, 73, 74, 75, 76, 77, 82, 83, 86, 89, 90, 91, 100, 101, 102, 105, 107, 108, 109, 146, 147, 148, 150, 153, 154, 155, 164, 165, 166, 172, 173, 178, 179, 180, 181, 182

Offset: 1

M. F. Hasler, May 05 2017

Cubefree means that there is no substring which is the repetition of three identical nonempty strings, see examples.

If n is not in the sequence, no number of the form n*2^k + m with 0 <= m < 2^k can be in the sequence, nor any number of the form m*2^k + n with 2^k > n, m >= 0.

			7 is not in the sequence, because 7 = 111[2] contains three consecutive "1"s.
8 is not in the sequence, because 8 = 1000[2] contains three consecutive "0"s.
More generally, no number congruent to 7 or congruent to 0 (mod 8) may be in the sequence.
Even more generally, no number of the form m*2^(k+3) +- n, n < 2^k, can be in this sequence.
42 is not in the sequence, because 42 = 101010[2] contains three consecutive "10"s.
From the comment follows that no number of the form 7*2^k, 8*2^k or 42*2^k may be in the sequence, for any k>=0. More generally, no number of the form 7*2^k + m, 8*2^k + m or 42*2^k + m may be in the sequence, for any 2^k > m >= 0.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A028445, A063037, A286261 (complement of this sequence).

isCubeFree:=proc(v) local n, L;
for n from 3 to nops(v) do for L to n/3 do
if v[n-L*2+1 .. n] = v[n-L*3+1 .. n-L] then RETURN(false) fi od od; true end;
a:=[];
for n from 1 to 512 do
if isCubeFree(convert(n, base, 2)) then a:=[op(a), n]; fi; od;
a;

from _future_ import division
def is_cubefree(s):
    l = len(s)
    for i in range(l-2):
        for j in range(1,(l-i)//3+1):
            if s[i:i+2*j] == s[i+j:i+3*j]:
                return False
    return True
A286262_list = [n for n in range(10**4) if is_cubefree(bin(n)[2:])] # Chai Wah Wu, May 06 2017

lim a(n)/n = infinity: sequence has asymptotic density 0.

A136037 Numbers with at least three adjacent equal digits in binary representation.

Original entry on oeis.org

7, 8, 14, 15, 16, 17, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 46, 47, 48, 49, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 78, 79, 80, 81, 87, 88, 92, 93, 94, 95, 96, 97, 98, 99, 103, 104, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

Reinhard Zumkeller, Dec 11 2007

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A007088.
Complement of A063037; union of A004779 and A004781.
Supersequence of A037970.

Select[Range[150],SequenceCount[IntegerDigits[#,2],{x_,x_,x_}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2020 *)

def ok(n): b = bin(n)[2:]; return "000" in b or "111" in b
print(list(filter(ok, range(120)))) # Michael S. Branicky, Jul 08 2021

A166535 Positive integers whose binary representation does not contain a run of more than three consecutive 0's or 1's.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

John W. Layman, Oct 16 2009

nonn

A179970 is a subsequence. - Reinhard Zumkeller, Aug 04 2010

There are A000073(n+2) terms with binary length n. - Rémy Sigrist, Sep 30 2022

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000073, A063037, A000975.

Select[Range[100],Max[Length/@Split[IntegerDigits[#,2]]]<4&] (* Harvey P. Dale, Aug 01 2020 *)

tribonacci(n) = ([0,1,0; 0,0,1; 1,1,1]^n)[2,1]
a(n) = { if (n<=4, return (n), my (s=1); for (i=1, oo, my (f=tribonacci(i+2)); i
f (nRémy Sigrist, Sep 30 2022

It appears (but has not yet been proved) that the terms of a(n) can be computed recursively as follows. Let {c(i)} be defined for i >= 5 by c(i)=2c(i-1)+1 if i is congruent to 2 mod 4, else c(i)=2c(i-1)-1. I.e., {c(i)}={1,3,5,9,17,35,...} for i=5,6,7,... . Let a(n)=n for n=1,2,...,7. For n>7, choose k so that s(k) < n < s(k+1), where s(k) = Sum_{j=3..k} t(j) with t(j) being the j-th term of the tribonacci sequence A000073 (with initial terms t(0)=0, t(1)=0, t(2)=1). Then a(n) = c(k) + 2a(s(k)) - a(2s(k)+1). This has been verified for n up to 2400.

{i: A043276(i) <= 3}. - R. J. Mathar, Jun 04 2021

A178905 Numbers without 3 consecutive equal digits in any base b >= 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 18, 19, 20, 22, 25, 36, 37, 38, 44, 45, 50, 51, 52, 74, 75, 76, 77, 89, 90, 100, 101, 102, 105, 109, 147, 150, 153, 154, 165, 166, 173, 178, 179, 180, 181, 204, 205, 210, 212, 214, 217, 293, 294, 299, 300, 301, 306, 308, 309, 329

Offset: 1

Joonas Pohjonen, Jun 22 2010

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A063037.

Prepend[Cases[Range[329], n_ /; NoneTrue[Range[2, (Sqrt[4 n - 3] - 1)/2], MatchQ[IntegerDigits[n, #], {_, d_, d_, d_, _}] &]], 0] (* Vladimir Reshetnikov, Mar 20 2022 *)

from sympy.ntheory.digits import digits
def three_in_a_row(s):
    return any(s[i] == s[i+1] == s[i+2] for i in range(len(s) - 2))
def ok(n):
    if n < 7: return True
    b = 2
    d = digits(n, b)[1:]
    while len(d) >= 3:
        if three_in_a_row(d): return False
        b += 1
        d = digits(n, b)[1:]
    return True
print([k for k in range(331) if ok(k)]) # Michael S. Branicky, Mar 27 2022

A371944 The binary expansion of a(n) corresponds to the ordinal transform (reduced modulo 2) of the binary expansion of n.

Original entry on oeis.org

0, 1, 3, 2, 6, 6, 5, 5, 13, 12, 12, 13, 10, 11, 11, 10, 26, 26, 25, 25, 25, 25, 26, 26, 21, 21, 22, 22, 22, 22, 21, 21, 53, 52, 52, 53, 50, 51, 51, 50, 50, 51, 51, 50, 53, 52, 52, 53, 42, 43, 43, 42, 45, 44, 44, 45, 45, 44, 44, 45, 42, 43, 43, 42, 106, 106

Offset: 0

Rémy Sigrist, Apr 13 2024

Leading zeros are ignored.

All terms belong to A063037.

			For n = 43: the binary expansion of 43 is "101011", the corresponding ordinal transform is "1, 1, 2, 2, 3, 4", reducing modulo 2 yields "110010", the binary expansion of a(43), so a(43) = 50.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
OEIS Wiki, Ordinal transform
Index entries for sequences related to binary expansion of n

Cf. A063037, A070939, A371961.

{0}~Join~Array[(c[0] = 1; c[1] = 1; FromDigits[Map[Mod[c[#]++, 2] &, IntegerDigits[#, 2] ], 2]) &, 120] (* Michael De Vlieger, Apr 16 2024 *)

a(n) = { my (b = binary(n), f = vector(2)); for (i = 1, #b, b[i] = f[1+b[i]]++;); fromdigits(b % 2, 2); }

A070939(a(n)) = A070939(n).

a(floor(n/2)) = floor(a(n)/2).

A201992 Numbers whose binary representations are found in the Thue-Morse sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 22, 25, 26, 37, 38, 41, 44, 45, 50, 51, 52, 75, 76, 77, 82, 83, 89, 90, 101, 102, 105, 150, 153, 154, 165, 166, 179, 180, 203, 205, 210, 211, 300, 301, 306, 308, 331, 332, 358, 361, 406, 410, 421, 422, 601

Offset: 0

Walt Rorie-Baety, Dec 07 2011

Interpreting A010060 as a bit string, this sequence contains the decimal equivalents of the subsequences, in order.

			The binary representation of 21 (10101) has an overlapping square sequence (1X1X1, where X is any binary sequence, in this case, X = 0), and so is not in the sequence. Compare to A063037.

Walt Rorie-Baety, Table of n, a(n) for n = 0..2500
Project Euler, Problem 361: Subsequence of Thue-Morse sequence

Cf. A010060, A063037.

a201992 = 0: concatMap (\n -> Set.toList . Set.fromList . map binRep . filter ((==[1]).take 1) . window n . take (n*2^n) $ a010060) [1..] where
  {window n = takeWhile (full . drop (n-1)) . map (take n) .  tails; binRep = foldl' (\a b -> 2*a+b) 0}; full = not . null

Module[{nn=10000,tm},tm=Table[ThueMorse[n],{n,0,nn}];Join[{0},Position[ Table[ If[SequenceCount[tm,IntegerDigits[k,2]]>0,1,0],{k,1000}], 1]]]// Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 03 2018 *)

Helper function added and name of value in program changed for better understanding by Walt Rorie-Baety, Mar 25 2012

cp's OEIS Frontend

A357522 Reverse run lengths in binary expansions of terms of A063037: for n >= 0, a(n) is the unique k such that A063037(1+k) = A056539(A063037(1+n)).

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

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

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A179888 Starting with a(1)=2: if m is a term then also 4*m+1 and 4*m+2.

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Maple

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A286262 Numbers whose binary expansion is a cubefree string.

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Maple

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A136037 Numbers with at least three adjacent equal digits in binary representation.

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A166535 Positive integers whose binary representation does not contain a run of more than three consecutive 0's or 1's.

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Mathematica

PARI

A179888 Starting with a(1)=2: if m is a term then also 4m+1 and 4m+2.