A057148 -id:A057148 - cp's OEIS frontend

A154810 Nonpalindromic numbers with binary digits only.

10, 100, 110, 1000, 1010, 1011, 1100, 1101, 1110, 10000, 10010, 10011, 10100, 10110, 10111, 11000, 11001, 11010, 11100, 11101, 11110, 100000, 100010, 100011, 100100, 100101, 100110, 100111, 101000, 101001, 101010, 101011, 101100, 101110, 101111

Offset: 1

Omar E. Pol, Jan 24 2009

A154809 written in base 2.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A007088, A029742, A057148, A154809.

Map[FromDigits, Select[IntegerDigits[Range[50], 2], !PalindromeQ[#] &]] (* Paolo Xausa, Jul 24 2024 *)

def A154810(n):
    def f(x): return n+(x>>(l:=x.bit_length())-(k:=l+1>>1))-(int(bin(x)[k+1:1:-1],2)>(x&(1<Chai Wah Wu, Jul 24 2024

a(n) = A007088(A154809(n)). - Michel Marcus, Jul 24 2024

Extended by Ray Chandler, Mar 14 2010

A192776 Squares of binary palindromes.

Original entry on oeis.org

0, 1, 1001, 11001, 110001, 1010001, 11100001, 100100001, 110111001, 1011011001, 1111000001, 10001000001, 11111101001, 101000101001, 111110000001, 1000010000001, 1010011010001, 1110000111001, 10000111001001, 10011001001001, 10110010111001, 11011101010001, 11111100000001, 100000100000001, 101101101110001, 110101001011001, 1000101110001001, 1001010010001001, 1011101101011001, 1101000001110001, 1111111000000001, 10000001000000001, 10010001100100001

Offset: 1

N. J. A. Sloane, Jul 09 2011

This is A192775 written in base 2, also A006995 (or A057148) squared. Cf. A006995, A057148.

def A192776(n):
    if n == 1: return 0
    a = 1<<(l:=n.bit_length()-2)
    m = a|(n&a-1)
    return int(bin(((m<Chai Wah Wu, Jul 23 2024

A262556 Even numbers that are not the sum of two binary palindromes (written in base 2).

Original entry on oeis.org

10110000, 10111100, 11010000, 11110010, 11110100, 100110110, 1000001100, 1001110100, 1010010000, 1011100000, 1011110010, 1100011010, 1101000000, 1101011110, 1101100010, 1101100100, 1101110000, 1110100100, 1110110000, 1111100010, 1111101000, 10010011100, 10011011000, 10100011100, 10100011110

Offset: 1

N. J. A. Sloane, Oct 12 2015

This is A261678 written in base 2.
It would be nice to have an independent characterization of these binary strings.
Binary palindromes are odd, so no odd number is the sum of two of them.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A261678, A241491.

See also A006995 and A057148 (binary palindromes).

A342036 Palindromes of even length only using 0 or 1.

Original entry on oeis.org

0, 11, 1001, 1111, 100001, 101101, 110011, 111111, 10000001, 10011001, 10100101, 10111101, 11000011, 11011011, 11100111, 11111111, 1000000001, 1000110001, 1001001001, 1001111001, 1010000101, 1010110101, 1011001101, 1011111101, 1100000011, 1100110011

Offset: 0

Seiichi Manyama, Feb 26 2021

Subsequence of A057148.

a(n) is a multiple of 11.

			A006995|A057148|A048701|A342036|A048700|A342040
-------+-------+-------+-------+-------+-------
     0 |     0 |     0 |     0 |       |
     1 |     1 |       |       |     1 |     1
     3 |    11 |     3 |    11 |       |
     5 |   101 |       |       |     5 |   101
     7 |   111 |       |       |     7 |   111
     9 |  1001 |     9 |  1001 |       |
    15 |  1111 |    15 |  1111 |       |
    17 | 10001 |       |       |    17 | 10001

Cf. A006995, A007088, A048701, A057148, A070939, A305989, A342040.

Array[FromDigits@ Join[#, Reverse[#]] &@ IntegerDigits[#, 2] &, 26, 0] (* Michael De Vlieger, Feb 26 2021 *)

def a(n): b = bin(n)[2:]; return int(b+b[::-1])
print([a(n) for n in range(27)]) # Michael S. Branicky, Feb 26 2021

def A(n)
  str = n.to_s(2)
  (str + str.reverse).to_i
end
def A342036(n)
  (0..n).map{|i| A(i)}
end
p A342036(30)

a(n) = A007088(n) * 10^A070939(n) + A305989(n).

a(n) = A007088(A048701(n)). - Michel Marcus, Feb 26 2021

Offset changed to 0 by Andrey Zabolotskiy, Dec 26 2022

A342040 Binary palindromes of odd length.

Original entry on oeis.org

1, 101, 111, 10001, 10101, 11011, 11111, 1000001, 1001001, 1010101, 1011101, 1100011, 1101011, 1110111, 1111111, 100000001, 100010001, 100101001, 100111001, 101000101, 101010101, 101101101, 101111101, 110000011, 110010011, 110101011

Offset: 1

Seiichi Manyama, Feb 26 2021

Subsequence of A057148.

Cf. A006995, A007088, A048700, A057148, A070939, A305989, A342036.

a[1] = 1; a[n_] := FromDigits[Join[#, {Mod[n, 2]}, Reverse[#]] &@ IntegerDigits[Floor[n/2], 2]]; Array[a, 26] (* Amiram Eldar, Apr 28 2021 *)

def A342040(n):
    s = bin(n)[2:]
    return int(s+s[-2::-1]) # Chai Wah Wu, Feb 26 2021

a(n) = A007088(A048700(n)).

A043261 Sum of the binary digits of the n-th base-2 palindrome.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 3, 4, 5, 2, 4, 4, 6, 2, 3, 4, 5, 4, 5, 6, 7, 2, 4, 4, 6, 4, 6, 6, 8, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 2, 4, 4, 6, 4, 6, 6, 8, 4, 6, 6, 8, 6, 8, 8, 10, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8

Offset: 1

Clark Kimberling

			The fourth base-2 palindrome is 5 = 101_2, so a(4) = 1+0+1 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006995 (base-2 palindromes), A057148.

b:= proc(n) option remember;
  procname(floor(n/2)) end proc;
b(1):= 0; b(2):= 1;
c:= proc(n) option remember;
  procname(floor(n/2)) + (n mod 2) end proc;
c(1):= 0; c(2):= 1;
A043261:= n -> 2*c(n) - (n mod 2)*b(n);
A043261(2):= 1;# Robert Israel, Apr 06 2014

def A043261(n):
    if n == 1: return 0
    a = 1<<(l:=n.bit_length()-2)
    m = a|(n&a-1)
    return (m.bit_count()<<1) - (0 if a&n else m&1) # Chai Wah Wu, Jun 13 2024

Let b(1) = 0, b(2) = 1, otherwise b(2*n-1) = b(n-1) and b(2*n) = b(n).
Let c(1) = 0, c(2) = 1, otherwise c(2*n-1) = c(n-1)+1 and c(2*n) = c(n).
Then for n >= 2, a(2*n-1) = 2*c(2*n-1) - b(2*n-1) and a(2*n) = 2*c(2*n).

edited by Robert Israel, Apr 06 2014

A143906 a(n) is A143905(n) written in binary.

Original entry on oeis.org

1001, 10011001, 10100101, 11000011, 100011110001, 100101101001, 100110011001, 101001100101, 101010010101, 101100001101, 110001100011, 110010010011, 110100001011, 111000000111, 1000011111100001, 1000101111010001

Offset: 1

Leroy Quet, Sep 04 2008

This sequence lists all finite binary strings that are palindromes with an equal number of 0's and 1's and that each start and end with 1, listed in order of the strings' numerical values if the strings are considered to be base 2 numbers.

Cf. A006995, A031443, A143905.

Cf. A057148, A071925.

Intersection of A057148 and A071925. - R. J. Mathar, Sep 05 2008

Extended by R. J. Mathar, Sep 05 2008

Corrected comment. - Leroy Quet, Oct 13 2008

A307319 Minimum number of concatenated palindromes needed to express first n terms of the Thue-Morse sequence (A010060).

Original entry on oeis.org

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

Offset: 0

Jeffrey Shallit, Apr 02 2019

nonn

			The first 6 terms of the Thue-Morse sequence are 011010, and this can be written as the concatenation of three palindromes: (0110)(1)(0), and no fewer.

Anna E. Frid, Prefix palindromic length of the Thue-Morse word, arXiv:1906.09392 [cs.DM], 2019. See p. 3.

Cf. A010060, A057148.

A327296 a(n) is the decimal value of the largest binary palindrome that can be obtained from the digits of the binary representation of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 5, 7, 1, 9, 9, 7, 9, 7, 7, 15, 1, 17, 17, 21, 17, 21, 21, 27, 17, 21, 21, 27, 21, 27, 27, 31, 1, 33, 33, 21, 33, 21, 21, 51, 33, 21, 21, 51, 21, 51, 51, 31, 33, 21, 21, 51, 21, 51, 51, 31, 21, 51, 51, 31, 51, 31, 31, 63, 1, 65, 65, 73, 65, 73

Offset: 0

Ian Band, Sep 16 2019

a(n) is the largest-valued binary palindrome that can be constructed from the binary representation of n. The palindrome may not have a leading 0.
When constructing a palindrome from a binary string, digits may only be permuted and deleted. I.e., the constructed palindrome will contain the same number or fewer 1's and 0's as the original binary string.
The algorithm is as follows:
If there is an odd number of 1's in the original binary string, start by placing a 1 in the middle of the output palindrome. Note that the remaining number of 1's is now even. If the remaining number of 1's is zero, then the constructed palindrome is simply "1" because the constructed palindrome cannot have leading zeros. If the remaining number of 1's is greater than zero, then place the remaining 0's evenly on either side of the previously placed "1". Because the output must be a palindrome, if the original string contains an odd number of 0's, then one of the 0's will be discarded. Lastly, place the remaining 1's on either end of the constructed palindrome.
If there is an even number of 1's in the original binary string, start by placing all of the 0's from the original string in the center of the palindrome, then place the 1's evenly on either side of the 0's.

			The largest palindrome that can be constructed:
For 101  it is  101, thus a(5)  = 5.
For 1000 it is    1, thus a(8)  = 1.
For 1010 it is 1001, thus a(10) = 9.
For 1011 it is  111, thus a(11) = 7.

Ian Band, Table of n, a(n) for n = 0..4094

Cf. A006995 (binary palindromes), A057148.

a[n_] := Block[{o,z}, {o,z} = DigitCount[n, 2]; If[o==1, 1, If[OddQ@ o, {o,z} = Floor[{o,z} /2]; 2^(z + o) + (2^o - 1) (1 + 2^(o + 1 + 2 z)), o = o/2; (2^o - 1) (1 + 2^(o + z))]]]; a /@ Range[0, 70] (* Giovanni Resta, Sep 16 2019 *)

def a(n):
    #convert n to binary string
    bin_str = str(bin(n))[2:]
    #count 1's and 0's in the string
    zeros = bin_str.count('0')
    ones  = len(bin_str) - zeros
    if(ones == 1 or ones == 0):
        return ones
    pal = ""
    if(ones % 2 == 1):
        #start by placing a '1' in the middle of the string
        pal = '1'
        #place as many 0's around the central '1' as possible
        for i in range(0, zeros >> 1):
                pal = '0' + pal + '0'
    else:
        #place all 0's in the center
        for i in range(0, zeros):
            pal += '0'
    #place the remaining 1's (guaranteed to be an even number, because one '1' was placed in the middle) on either side of the palindrome
    for i in range(0, ones >> 1):
        pal = '1' + pal + '1'
    #return integer value of the constructed palindrome
    return int(pal, 2)

More terms from Giovanni Resta, Sep 16 2019

A330022 Length of shortest binary string containing, as contiguous blocks, all palindromes of length n.

Original entry on oeis.org

0, 2, 4, 8, 12, 22, 32

Offset: 0

Jeffrey Shallit, Nov 27 2019

Greedy supersequence algorithms give the upper bounds a(7) <= 60, a(8) <= 74, a(9) <= 142, a(10) <= 180, a(11) <= 344, a(12) <= 410, a(13) <= 798. Probably some of these are tight. The value for a(6) was computed by checking all 8! arrangements of the 8 palindromes of length 3, optimizing overlaps. Probably someone with more computing power could compute a(7) (resp., a(8)) by checking all 16! = 20922789888000 arrangements of the palindromes of length 7 (resp., 8).

			The corresponding strings for 1 <= n <= 6 are:
1: 01
2: 0011
3: 00010111
4: 000011001111
5: 0000010001010111011111
6: 00000011001111000010010110111111

Cf. A057148.

cp's OEIS Frontend

A154810 Nonpalindromic numbers with binary digits only.

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A192776 Squares of binary palindromes.

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A262556 Even numbers that are not the sum of two binary palindromes (written in base 2).

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A342036 Palindromes of even length only using 0 or 1.

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A342040 Binary palindromes of odd length.

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A043261 Sum of the binary digits of the n-th base-2 palindrome.

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A143906 a(n) is A143905(n) written in binary.

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A307319 Minimum number of concatenated palindromes needed to express first n terms of the Thue-Morse sequence (A010060).

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A327296 a(n) is the decimal value of the largest binary palindrome that can be obtained from the digits of the binary representation of n.

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