Jocasta Numbers
Sequential Construction of Jocasta Binaries
N = 1
B = Ø
N = 2
B = 0
B = 1
N = 3
B = 00 : Ø
B = 11 : Ø
N = 4
B = 000 : 0
B = 010 : 1
B = 101 : -1
B = 110 : 1
B = 111 : 0
N = 5
B = 0000 : 00
B = 0110 : 11
B = 1001 : -11
B = 1110 : 11
B = 1111 : 00
N = 6
B = 00000 : 000
B = 00100 : 010
B = 01010 : 101
B = 01100 : 110
B = 01110 : 111
B = 10001 : -111
B = 10011 : -110
B = 10101 : -101
B = 11010 : 101
B = 11011 : -010
B = 11100 : 010 | 110
B = 11110 : 111
B = 11111 : 000
N = 7
B = 000000 : 0000
B = 001100 : 0110
B = 010010 : 1001
B = 011100 : 1110
B = 011110 : 1111
B = 100001 : -1111
B = 100011 : -1110
B = 101101 : -1001
B = 110010 : 1001
B = 110011 : -0110
B = 111100 : 0110 | 1110
B = 111110 : 1111
B = 111111 : 0000
N = 8
B = 0000000 : 00000
B = 0001000 : 00100
B = 0010100 : 01010
B = 0011000 : 01100
B = 0011100 : 01110
B = 0100010 : 10001
B = 0100110 : 10011
B = 0101010 : 10101
B = 0110100 : 11010
B = 0110110 : 11011
B = 0111000 : 11100
B = 0111100 : 11110
B = 0111110 : 11111
B = 1000001 : -11111
B = 1000011 : -11110
B = 1000111 : -11100
B = 1001001 : -11011
B = 1001011 : -11010
B = 1010101 : -10101
B = 1011001 : -10011
B = 1011101 : -10001
B = 1100010 : 10001
B = 1100011 : -01110
B = 1100110 : 10011
B = 1100111 : -01100
B = 1101010 : 10101
B = 1101011 : -01010
B = 1110100 : 01010 | 11010
B = 1110110 : 11011
B = 1110111 : -00100
B = 1111000 : 00100 | 01100 | 11100
B = 1111100 : 01110 | 11110
B = 1111110 : 11111
B = 1111111 : 00000
Full Construction for N=8
| + | + | + | + | − | − | − | − | 1111000 |
| + | + | + | = | = | − | − | − | 1111100 |
| + | + | + | − | + | − | − | − | 1110100 |
| + | + | = | + | − | = | − | − | 1111000 |
| + | + | = | = | = | = | − | − | 1111110 |
| + | + | = | − | + | = | − | − | 1110110 |
| + | + | − | + | − | + | − | − | 1101010 |
| + | + | − | = | = | + | − | − | 1100010 |
| + | + | − | − | + | + | − | − | 1100110 |
| + | = | + | + | − | − | = | − | 1111000 |
| + | = | + | = | = | − | = | − | 1111100 |
| + | = | + | − | + | − | = | − | 1110100 |
| + | = | = | + | − | = | = | − | 1111000 |
| + | = | = | = | = | = | = | − | 1111111 |
| + | = | = | − | + | = | = | − | 1110111 |
| + | = | − | + | − | + | = | − | 1101011 |
| + | = | − | = | = | + | = | − | 1100011 |
| + | = | − | − | + | + | = | − | 1100111 |
| + | − | + | + | − | − | + | − | 1011001 |
| + | − | + | = | = | − | + | − | 1011101 |
| + | − | + | − | + | − | + | − | 1010101 |
| + | − | = | + | − | = | + | − | 1001001 |
| + | − | = | = | = | = | + | − | 1000001 |
| + | − | = | − | + | = | + | − | 1000111 |
| + | − | − | + | − | + | + | − | 1001011 |
| + | − | − | = | = | + | + | − | 1000011 |
| + | − | − | − | + | + | + | − | 1000111 |
| = | + | + | + | − | − | − | = | 0111000 |
| = | + | + | = | = | − | − | = | 0111100 |
| = | + | + | − | + | − | − | = | 0110100 |
| = | + | = | + | − | = | − | = | 0111000 |
| = | + | = | = | = | = | − | = | 0111110 |
| = | + | = | − | + | = | − | = | 0110110 |
| = | + | − | + | − | + | − | = | 0101010 |
| = | + | − | = | = | + | − | = | 0100010 |
| = | + | − | − | + | + | − | = | 0100110 |
| = | = | + | + | − | − | = | = | 0011000 |
| = | = | + | = | = | − | = | = | 0011100 |
| = | = | + | − | + | − | = | = | 0010100 |
| = | = | = | + | − | = | = | = | 0001000 |
| = | = | = | = | = | = | = | = | 0000000 |
Details
N = 1
N = 2
N = 3
| X | + | = | − |
| J- | -1 | b-1 | b |
| J | b-1 | 2b-2 | b-1 |
| B | | 1 | 1 |
N = 4
| X | + | + | − | − |
| J- | 0 | -1 | b-1 | b |
| J | b | b-2 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 0 |
| X | + | = | = | − |
| J- | -1 | b-1 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | b-1 |
| B | | 1 | 1 | 1 |
| X | + | − | + | − |
| J- | -1 | b | -1 | b |
| J | b-1 | b-1 | b-1 | b-1 |
| B | | 1 | 0 | 1 |
| X | = | + | − | = |
| J- | 0 | -1 | b | 0 |
| J | 0 | b-1 | b-1 | 0 |
| B | | 0 | 1 | 0 |
N = 5
| X | + | + | = | − | − |
| J- | 0 | -1 | b-1 | b-1 | b |
| J | b | b-2 | 2b-2 | b-2 | b |
| = | b-1 | 2b-2 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 1 | 0 |
| X | + | = | = | = | − |
| J- | -1 | b-1 | b-1 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 |
| B | | 1 | 1 | 1 | 1 |
| X | + | − | = | + | − |
| J- | -1 | b | 0 | -1 | b |
| J | b-1 | b-1 | 0 | b-1 | b-1 |
| B | | 1 | 0 | 0 | 1 |
| X | = | + | = | − | = |
| J- | 0 | -1 | b-1 | b | 0 |
| J | 0 | b-1 | 2b-2 | b-1 | 0 |
| B | | 0 | 1 | 1 | 0 |
N = 6
Considering only cases of the form +****−
| X | + | + | + | − | − | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 0 | 0 |
| X | + | + | = | = | − | − |
| J- | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-2 | 2b-2 | 2b-2 | b-2 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 |
| X | + | + | − | + | − | − |
| J- | 0 | -1 | b | -1 | b-1 | b |
| J | b | b-2 | b-1 | b-1 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | b-1 | b-1 | 0 |
| B | | 1 | 1 | 0 | 1 | 0 |
| X | + | = | + | − | = | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 0 | 0 |
| X | + | = | = | = | = | − |
| J- | -1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 |
| B | | 1 | 1 | 1 | 1 | 1 |
| X | + | = | − | + | = | − |
| J- | -1 | b-1 | b | -1 | b-1 | b |
| J | b-1 | 2b-2 | b-1 | b-1 | 2b-2 | b-1 |
| B | | 1 | 1 | 0 | 1 | 1 |
| X | + | − | + | − | + | − |
| J- | -1 | b | -1 | b | -1 | b |
| J | b-1 | b-1 | b-1 | b-1 | b-1 | b-1 |
| B | | 1 | 0 | 1 | 0 | 1 |
| X | + | − | = | = | + | − |
| J- | -1 | b | 0 | 0 | -1 | b |
| J | b-1 | b-1 | 0 | 0 | b-1 | b-1 |
| B | | 1 | 0 | 0 | 0 | 1 |
| X | + | − | − | + | + | − |
| J- | -1 | b-1 | b | 0 | -1 | b |
| J | b-1 | b-2 | b | b | b-2 | b-1 |
| = | b-1 | b-1 | 0 | b-1 | 2b-2 | b-1 |
| B | | 1 | 0 | 0 | 1 | 1 |
N = 7
Considering only cases of the form +*****−
| X | + | + | + | = | − | − | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | 2b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 |
| X | + | + | = | = | = | − | − |
| J- | 0 | -1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b | b-2 | 2b-2 | 2b-2 | 2b-2 | b-2 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 1 | 1 | 1 | 0 |
| X | + | + | − | = | + | − | − |
| J- | 0 | -1 | b | 0 | -1 | b-1 | b |
| J | b | b-2 | b-1 | 0 | b-1 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | 0 | b-1 | b-1 | 0 |
| B | | 1 | 1 | 0 | 0 | 1 | 0 |
| X | + | = | + | = | − | = | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | 2b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 |
| X | + | = | = | = | = | = | − |
| J- | -1 | b-1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 |
| B | | 1 | 1 | 1 | 1 | 1 | 1 |
| X | + | = | − | = | + | = | − |
| J- | -1 | b-1 | b | 0 | -1 | b-1 | b |
| J | b-1 | 2b-2 | b-1 | 0 | b-1 | 2b-2 | b-1 |
| B | | 1 | 1 | 0 | 0 | 1 | 1 |
| X | + | − | + | = | − | + | − |
| J- | -1 | b | -1 | b-1 | b | -1 | b |
| J | b-1 | b-1 | b-1 | 2b-2 | b-1 | b-1 | b-1 |
| B | | 1 | 0 | 1 | 1 | 0 | 1 |
| X | + | − | = | = | = | + | − |
| J- | -1 | b | 0 | 0 | 0 | -1 | b |
| J | b-1 | b-1 | 0 | 0 | 0 | b-1 | b-1 |
| B | | 1 | 0 | 0 | 0 | 0 | 1 |
| X | + | − | − | = | + | + | − |
| J- | -1 | b-1 | b | 0 | 0 | -1 | b |
| J | b-1 | b-2 | b | 0 | b | b-2 | b-1 |
| = | b-1 | b-1 | 0 | 0 | b-1 | 2b-2 | b-1 |
| B | | 1 | 0 | 0 | 0 | 1 | 1 |
N = 8
Considering only cases of the form +******− where ****** ≥ R(******)
| X | + | + | + | + | − | − | − | − |
| J- | 0 | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-1 | b-2 | b-2 | b-1 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| X | + | + | + | = | = | − | − | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | 2b-2 | 2b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| X | + | + | + | − | + | − | − | − |
| J- | 0 | 0 | -1 | b | -1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | b-1 | b-1 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | b-1 | b-1 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
| X | + | + | = | + | − | = | − | − |
| J- | 0 | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-1 | b-2 | b-2 | b-1 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| X | + | + | = | = | = | = | − | − |
| J- | 0 | -1 | b-1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b | b-2 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-2 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
| X | + | + | = | − | + | = | − | − |
| J- | 0 | -1 | b-1 | b | -1 | b-1 | b-1 | b |
| J | b | b-2 | 2b-2 | b-1 | b-1 | 2b-2 | b-2 | b |
| = | b-1 | 2b-2 | 2b-2 | b-1 | b-1 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 1 | 0 | 1 | 1 | 0 |
| X | + | + | − | + | − | + | − | − |
| J- | 0 | -1 | b | -1 | b | -1 | b-1 | b |
| J | b | b-2 | b-1 | b-1 | b-1 | b-1 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | b-1 | b-1 | b-1 | b-1 | 0 |
| B | | 1 | 1 | 0 | 1 | 0 | 1 | 0 |
| X | + | + | − | = | = | + | − | − |
| J- | 0 | -1 | b | 0 | 0 | -1 | b-1 | b |
| J | b | b-2 | b-1 | 0 | 0 | b-1 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | 0 | 0 | b-1 | b-1 | 0 |
| B | | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
| X | + | + | − | − | + | + | − | − |
| J- | 0 | -1 | b-1 | b | 0 | -1 | b-1 | b |
| J | b | b-2 | b-2 | b | b | b-2 | b-2 | b |
| = | b-1 | 2b-2 | b-1 | 0 | b-1 | 2b-2 | b-1 | 0 |
| B | | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| X | + | = | + | + | − | − | = | − |
| J- | 0 | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-1 | b-2 | b-2 | b-1 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| X | + | = | + | = | = | − | = | − |
| J- | 0 | 0 | -1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | 2b-2 | 2b-2 | b-2 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| X | + | = | + | − | + | − | = | − |
| J- | 0 | 0 | -1 | b | -1 | b-1 | b-1 | b |
| J | b | b-1 | b-2 | b-1 | b-1 | b-2 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | b-1 | b-1 | b-1 | 0 | 0 |
| B | | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
| X | + | = | = | + | − | = | = | − |
| J- | 0 | 0 | 0 | -1 | b-1 | b-1 | b-1 | b |
| J | b | b-1 | b-1 | b-2 | b-2 | b-1 | b-1 | b |
| = | b-1 | 2b-2 | 2b-2 | 2b-2 | b-1 | 0 | 0 | 0 |
| B | | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| X | + | = | = | = | = | = | = | − |
| J- | -1 | b-1 | b-1 | b-1 | b-1 | b-1 | b-1 | b |
| J | b-1 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | 2b-2 | b-1 |
| B | | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
