Identifier
- St001413: Binary words ⟶ ℤ
Values
=>
0=>0
1=>0
00=>1
01=>0
10=>0
11=>1
000=>1
001=>1
010=>0
011=>0
100=>0
101=>0
110=>1
111=>1
0000=>2
0001=>1
0010=>1
0011=>1
0100=>0
0101=>0
0110=>2
0111=>0
1000=>0
1001=>2
1010=>0
1011=>0
1100=>1
1101=>1
1110=>1
1111=>2
00000=>2
00001=>2
00010=>1
00011=>1
00100=>1
00101=>1
00110=>1
00111=>1
01000=>0
01001=>0
01010=>0
01011=>0
01100=>2
01101=>2
01110=>0
01111=>0
10000=>0
10001=>0
10010=>2
10011=>2
10100=>0
10101=>0
10110=>0
10111=>0
11000=>1
11001=>1
11010=>1
11011=>1
11100=>1
11101=>1
11110=>2
11111=>2
000000=>3
000001=>2
000010=>2
000011=>2
000100=>1
000101=>1
000110=>1
000111=>1
001000=>1
001001=>1
001010=>1
001011=>1
001100=>3
001101=>1
001110=>1
001111=>1
010000=>0
010001=>0
010010=>3
010011=>0
010100=>0
010101=>0
010110=>0
010111=>0
011000=>2
011001=>2
011010=>2
011011=>2
011100=>0
011101=>0
011110=>3
011111=>0
100000=>0
100001=>3
100010=>0
100011=>0
100100=>2
100101=>2
100110=>2
100111=>2
101000=>0
101001=>0
101010=>0
101011=>0
101100=>0
101101=>3
101110=>0
101111=>0
110000=>1
110001=>1
110010=>1
110011=>3
110100=>1
110101=>1
110110=>1
110111=>1
111000=>1
111001=>1
111010=>1
111011=>1
111100=>2
111101=>2
111110=>2
111111=>3
0000000=>3
0000001=>3
0000010=>2
0000011=>2
0000100=>2
0000101=>2
0000110=>2
0000111=>2
0001000=>1
0001001=>1
0001010=>1
0001011=>1
0001100=>1
0001101=>1
0001110=>1
0001111=>1
0010000=>1
0010001=>1
0010010=>1
0010011=>1
0010100=>1
0010101=>1
0010110=>1
0010111=>1
0011000=>3
0011001=>3
0011010=>1
0011011=>1
0011100=>1
0011101=>1
0011110=>1
0011111=>1
0100000=>0
0100001=>0
0100010=>0
0100011=>0
0100100=>3
0100101=>3
0100110=>0
0100111=>0
0101000=>0
0101001=>0
0101010=>0
0101011=>0
0101100=>0
0101101=>0
0101110=>0
0101111=>0
0110000=>2
0110001=>2
0110010=>2
0110011=>2
0110100=>2
0110101=>2
0110110=>2
0110111=>2
0111000=>0
0111001=>0
0111010=>0
0111011=>0
0111100=>3
0111101=>3
0111110=>0
0111111=>0
1000000=>0
1000001=>0
1000010=>3
1000011=>3
1000100=>0
1000101=>0
1000110=>0
1000111=>0
1001000=>2
1001001=>2
1001010=>2
1001011=>2
1001100=>2
1001101=>2
1001110=>2
1001111=>2
1010000=>0
1010001=>0
1010010=>0
1010011=>0
1010100=>0
1010101=>0
1010110=>0
1010111=>0
1011000=>0
1011001=>0
1011010=>3
1011011=>3
1011100=>0
1011101=>0
1011110=>0
1011111=>0
1100000=>1
1100001=>1
1100010=>1
1100011=>1
1100100=>1
1100101=>1
1100110=>3
1100111=>3
1101000=>1
1101001=>1
1101010=>1
1101011=>1
1101100=>1
1101101=>1
1101110=>1
1101111=>1
1110000=>1
1110001=>1
1110010=>1
1110011=>1
1110100=>1
1110101=>1
1110110=>1
1110111=>1
1111000=>2
1111001=>2
1111010=>2
1111011=>2
1111100=>2
1111101=>2
1111110=>3
1111111=>3
00000000=>4
00000001=>3
00000010=>3
00000011=>3
00000100=>2
00000101=>2
00000110=>2
00000111=>2
00001000=>2
00001001=>2
00001010=>2
00001011=>2
00001100=>2
00001101=>2
00001110=>2
00001111=>2
00010000=>1
00010001=>1
00010010=>1
00010011=>1
00010100=>1
00010101=>1
00010110=>1
00010111=>1
00011000=>4
00011001=>1
00011010=>1
00011011=>1
00011100=>1
00011101=>1
00011110=>1
00011111=>1
00100000=>1
00100001=>1
00100010=>1
00100011=>1
00100100=>4
00100101=>1
00100110=>1
00100111=>1
00101000=>1
00101001=>1
00101010=>1
00101011=>1
00101100=>1
00101101=>1
00101110=>1
00101111=>1
00110000=>3
00110001=>3
00110010=>3
00110011=>3
00110100=>1
00110101=>1
00110110=>1
00110111=>1
00111000=>1
00111001=>1
00111010=>1
00111011=>1
00111100=>4
00111101=>1
00111110=>1
00111111=>1
01000000=>0
01000001=>0
01000010=>4
01000011=>0
01000100=>0
01000101=>0
01000110=>0
01000111=>0
01001000=>3
01001001=>3
01001010=>3
01001011=>3
01001100=>0
01001101=>0
01001110=>0
01001111=>0
01010000=>0
01010001=>0
01010010=>0
01010011=>0
01010100=>0
01010101=>0
01010110=>0
01010111=>0
01011000=>0
01011001=>0
01011010=>4
01011011=>0
01011100=>0
01011101=>0
01011110=>0
01011111=>0
01100000=>2
01100001=>2
01100010=>2
01100011=>2
01100100=>2
01100101=>2
01100110=>4
01100111=>2
01101000=>2
01101001=>2
01101010=>2
01101011=>2
01101100=>2
01101101=>2
01101110=>2
01101111=>2
01110000=>0
01110001=>0
01110010=>0
01110011=>0
01110100=>0
01110101=>0
01110110=>0
01110111=>0
01111000=>3
01111001=>3
01111010=>3
01111011=>3
01111100=>0
01111101=>0
01111110=>4
01111111=>0
10000000=>0
10000001=>4
10000010=>0
10000011=>0
10000100=>3
10000101=>3
10000110=>3
10000111=>3
10001000=>0
10001001=>0
10001010=>0
10001011=>0
10001100=>0
10001101=>0
10001110=>0
10001111=>0
10010000=>2
10010001=>2
10010010=>2
10010011=>2
10010100=>2
10010101=>2
10010110=>2
10010111=>2
10011000=>2
10011001=>4
10011010=>2
10011011=>2
10011100=>2
10011101=>2
10011110=>2
10011111=>2
10100000=>0
10100001=>0
10100010=>0
10100011=>0
10100100=>0
10100101=>4
10100110=>0
10100111=>0
10101000=>0
10101001=>0
10101010=>0
10101011=>0
10101100=>0
10101101=>0
10101110=>0
10101111=>0
10110000=>0
10110001=>0
10110010=>0
10110011=>0
10110100=>3
10110101=>3
10110110=>3
10110111=>3
10111000=>0
10111001=>0
10111010=>0
10111011=>0
10111100=>0
10111101=>4
10111110=>0
10111111=>0
11000000=>1
11000001=>1
11000010=>1
11000011=>4
11000100=>1
11000101=>1
11000110=>1
11000111=>1
11001000=>1
11001001=>1
11001010=>1
11001011=>1
11001100=>3
11001101=>3
11001110=>3
11001111=>3
11010000=>1
11010001=>1
11010010=>1
11010011=>1
11010100=>1
11010101=>1
11010110=>1
11010111=>1
11011000=>1
11011001=>1
11011010=>1
11011011=>4
11011100=>1
11011101=>1
11011110=>1
11011111=>1
11100000=>1
11100001=>1
11100010=>1
11100011=>1
11100100=>1
11100101=>1
11100110=>1
11100111=>4
11101000=>1
11101001=>1
11101010=>1
11101011=>1
11101100=>1
11101101=>1
11101110=>1
11101111=>1
11110000=>2
11110001=>2
11110010=>2
11110011=>2
11110100=>2
11110101=>2
11110110=>2
11110111=>2
11111000=>2
11111001=>2
11111010=>2
11111011=>2
11111100=>3
11111101=>3
11111110=>3
11111111=>4
000000000=>4
000000001=>4
000000010=>3
000000011=>3
000000100=>3
000000101=>3
000000110=>3
000000111=>3
000001000=>2
000001001=>2
000001010=>2
000001011=>2
000001100=>2
000001101=>2
000001110=>2
000001111=>2
000010000=>2
000010001=>2
000010010=>2
000010011=>2
000010100=>2
000010101=>2
000010110=>2
000010111=>2
000011000=>2
000011001=>2
000011010=>2
000011011=>2
000011100=>2
000011101=>2
000011110=>2
000011111=>2
000100000=>1
000100001=>1
000100010=>1
000100011=>1
000100100=>1
000100101=>1
000100110=>1
000100111=>1
000101000=>1
000101001=>1
000101010=>1
000101011=>1
000101100=>1
000101101=>1
000101110=>1
000101111=>1
000110000=>4
000110001=>4
000110010=>1
000110011=>1
000110100=>1
000110101=>1
000110110=>1
000110111=>1
000111000=>1
000111001=>1
000111010=>1
000111011=>1
000111100=>1
000111101=>1
000111110=>1
000111111=>1
001000000=>1
001000001=>1
001000010=>1
001000011=>1
001000100=>1
001000101=>1
001000110=>1
001000111=>1
001001000=>4
001001001=>4
001001010=>1
001001011=>1
001001100=>1
001001101=>1
001001110=>1
001001111=>1
001010000=>1
001010001=>1
001010010=>1
001010011=>1
001010100=>1
001010101=>1
001010110=>1
001010111=>1
001011000=>1
001011001=>1
001011010=>1
001011011=>1
001011100=>1
001011101=>1
001011110=>1
001011111=>1
001100000=>3
001100001=>3
001100010=>3
001100011=>3
001100100=>3
001100101=>3
001100110=>3
001100111=>3
001101000=>1
001101001=>1
001101010=>1
001101011=>1
001101100=>1
001101101=>1
001101110=>1
001101111=>1
001110000=>1
001110001=>1
001110010=>1
001110011=>1
001110100=>1
001110101=>1
001110110=>1
001110111=>1
001111000=>4
001111001=>4
001111010=>1
001111011=>1
001111100=>1
001111101=>1
001111110=>1
001111111=>1
010000000=>0
010000001=>0
010000010=>0
010000011=>0
010000100=>4
010000101=>4
010000110=>0
010000111=>0
010001000=>0
010001001=>0
010001010=>0
010001011=>0
010001100=>0
010001101=>0
010001110=>0
010001111=>0
010010000=>3
010010001=>3
010010010=>3
010010011=>3
010010100=>3
010010101=>3
010010110=>3
010010111=>3
010011000=>0
010011001=>0
010011010=>0
010011011=>0
010011100=>0
010011101=>0
010011110=>0
010011111=>0
010100000=>0
010100001=>0
010100010=>0
010100011=>0
010100100=>0
010100101=>0
010100110=>0
010100111=>0
010101000=>0
010101001=>0
010101010=>0
010101011=>0
010101100=>0
010101101=>0
010101110=>0
010101111=>0
010110000=>0
010110001=>0
010110010=>0
010110011=>0
010110100=>4
010110101=>4
010110110=>0
010110111=>0
010111000=>0
010111001=>0
010111010=>0
010111011=>0
010111100=>0
010111101=>0
010111110=>0
010111111=>0
011000000=>2
011000001=>2
011000010=>2
011000011=>2
011000100=>2
011000101=>2
011000110=>2
011000111=>2
011001000=>2
011001001=>2
011001010=>2
011001011=>2
011001100=>4
011001101=>4
011001110=>2
011001111=>2
011010000=>2
011010001=>2
011010010=>2
011010011=>2
011010100=>2
011010101=>2
011010110=>2
011010111=>2
011011000=>2
011011001=>2
011011010=>2
011011011=>2
011011100=>2
011011101=>2
011011110=>2
011011111=>2
011100000=>0
011100001=>0
011100010=>0
011100011=>0
011100100=>0
011100101=>0
011100110=>0
011100111=>0
011101000=>0
011101001=>0
011101010=>0
011101011=>0
011101100=>0
011101101=>0
011101110=>0
011101111=>0
011110000=>3
011110001=>3
011110010=>3
011110011=>3
011110100=>3
011110101=>3
011110110=>3
011110111=>3
011111000=>0
011111001=>0
011111010=>0
011111011=>0
011111100=>4
011111101=>4
011111110=>0
011111111=>0
100000000=>0
100000001=>0
100000010=>4
100000011=>4
100000100=>0
100000101=>0
100000110=>0
100000111=>0
100001000=>3
100001001=>3
100001010=>3
100001011=>3
100001100=>3
100001101=>3
100001110=>3
100001111=>3
100010000=>0
100010001=>0
100010010=>0
100010011=>0
100010100=>0
100010101=>0
100010110=>0
100010111=>0
100011000=>0
100011001=>0
100011010=>0
100011011=>0
100011100=>0
100011101=>0
100011110=>0
100011111=>0
100100000=>2
100100001=>2
100100010=>2
100100011=>2
100100100=>2
100100101=>2
100100110=>2
100100111=>2
100101000=>2
100101001=>2
100101010=>2
100101011=>2
100101100=>2
100101101=>2
100101110=>2
100101111=>2
100110000=>2
100110001=>2
100110010=>4
100110011=>4
100110100=>2
100110101=>2
100110110=>2
100110111=>2
100111000=>2
100111001=>2
100111010=>2
100111011=>2
100111100=>2
100111101=>2
100111110=>2
100111111=>2
101000000=>0
101000001=>0
101000010=>0
101000011=>0
101000100=>0
101000101=>0
101000110=>0
101000111=>0
101001000=>0
101001001=>0
101001010=>4
101001011=>4
101001100=>0
101001101=>0
101001110=>0
101001111=>0
101010000=>0
101010001=>0
101010010=>0
101010011=>0
101010100=>0
101010101=>0
101010110=>0
101010111=>0
101011000=>0
101011001=>0
101011010=>0
101011011=>0
101011100=>0
101011101=>0
101011110=>0
101011111=>0
101100000=>0
101100001=>0
101100010=>0
101100011=>0
101100100=>0
101100101=>0
101100110=>0
101100111=>0
101101000=>3
101101001=>3
101101010=>3
101101011=>3
101101100=>3
101101101=>3
101101110=>3
101101111=>3
101110000=>0
101110001=>0
101110010=>0
101110011=>0
101110100=>0
101110101=>0
101110110=>0
101110111=>0
101111000=>0
101111001=>0
101111010=>4
101111011=>4
101111100=>0
101111101=>0
101111110=>0
101111111=>0
110000000=>1
110000001=>1
110000010=>1
110000011=>1
110000100=>1
110000101=>1
110000110=>4
110000111=>4
110001000=>1
110001001=>1
110001010=>1
110001011=>1
110001100=>1
110001101=>1
110001110=>1
110001111=>1
110010000=>1
110010001=>1
110010010=>1
110010011=>1
110010100=>1
110010101=>1
110010110=>1
110010111=>1
110011000=>3
110011001=>3
110011010=>3
110011011=>3
110011100=>3
110011101=>3
110011110=>3
110011111=>3
110100000=>1
110100001=>1
110100010=>1
110100011=>1
110100100=>1
110100101=>1
110100110=>1
110100111=>1
110101000=>1
110101001=>1
110101010=>1
110101011=>1
110101100=>1
110101101=>1
110101110=>1
110101111=>1
110110000=>1
110110001=>1
110110010=>1
110110011=>1
110110100=>1
110110101=>1
110110110=>4
110110111=>4
110111000=>1
110111001=>1
110111010=>1
110111011=>1
110111100=>1
110111101=>1
110111110=>1
110111111=>1
111000000=>1
111000001=>1
111000010=>1
111000011=>1
111000100=>1
111000101=>1
111000110=>1
111000111=>1
111001000=>1
111001001=>1
111001010=>1
111001011=>1
111001100=>1
111001101=>1
111001110=>4
111001111=>4
111010000=>1
111010001=>1
111010010=>1
111010011=>1
111010100=>1
111010101=>1
111010110=>1
111010111=>1
111011000=>1
111011001=>1
111011010=>1
111011011=>1
111011100=>1
111011101=>1
111011110=>1
111011111=>1
111100000=>2
111100001=>2
111100010=>2
111100011=>2
111100100=>2
111100101=>2
111100110=>2
111100111=>2
111101000=>2
111101001=>2
111101010=>2
111101011=>2
111101100=>2
111101101=>2
111101110=>2
111101111=>2
111110000=>2
111110001=>2
111110010=>2
111110011=>2
111110100=>2
111110101=>2
111110110=>2
111110111=>2
111111000=>3
111111001=>3
111111010=>3
111111011=>3
111111100=>3
111111101=>3
111111110=>4
111111111=>4
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Description
Half the length of the longest even length palindromic prefix of a binary word.
References
[1] Gabric, D., Shallit, J. Borders, Palindrome Prefixes, and Square Prefixes arXiv:1906.03689
Code
def statistic(w): return max(k//2 for k in range(0,len(w)+1,2) if w[:k] == w[:k][::-1])
Created
Jun 11, 2019 at 07:24 by Martin Rubey
Updated
Mar 05, 2020 at 13:41 by Martin Rubey
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