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Observations on column 3 of TM #4 sequence, and mystery remains

For the TM #4 , hypothetically, an infinite
sequence is generated a bit at a time.

By arranging this sequence into rows of
eight, some simplication occurs:
(a) The first column is all 3s
(b) The second column is all 1s.
(c) And so on.

Suppose we denote the sequence by
s(1), s(2), s(3), s(4), and so on.

Then, the first column consists of
s(1), s(9), s(17), s(25), …

We say that the index takes on the values
1, 9, 17, 25, …
for the first column.

For the 3rd column, the index “j” takes on
the values 3, 11, 19, etc.

It appears that s(j) <= j+2 for indices j
with j == 3 (modulo 8), in the 3rd column.

I’ve tested in part the supposition that
s(j) can be obtained from j+2 by deleting
some of the leading bits of j+2 expressed in
binary. The number of bits deleted n_j can be
0, 1, … but there must result s(j) >= 5.

When s(j) = j+2 [3rd colum case still],
then we say that j+2 is “an allowable number”.

Is s( s(j) – 2) = s(j) ?

Are the s(j) allowable numbers?
Perhaps, yes.

Is s(j) the largest allowable number
contained in j+2 as a bit-string,
starting from the right, i.e. the least
significant bit?

Perhaps yes…

How do we determine, given j,
whether s(j) = j+2? When is
a number allowable?

This looks quite hard.

Below, for each j in
{3, 11, 19, …}
we show j+2 in the left column, and
s(j) in the right column:
j+2 s(j)
==================
101 101
1101 1101
10101 10101
11101 11101
100101 100101
101101 101101
110101 110101
111101 111101
1000101 101
1001101 1001101
1010101 1010101
1011101 1011101
1100101 1100101
1101101 1101101
1110101 1110101
1111101 1111101
10000101 101
10001101 1101
10010101 10010101
10011101 11101
10100101 100101
10101101 10101101
10110101 10110101
10111101 10111101
11000101 101
11001101 11001101
11010101 11010101
11011101 11011101
11100101 11100101
11101101 11101101
11110101 11110101
11111101 11111101
100000101 101
100001101 1101
100010101 10101
100011101 11101
100100101 100101
100101101 101101
100110101 100110101
100111101 111101
101000101 101
101001101 1001101
101010101 101010101
101011101 1011101
101100101 1100101
101101101 101101101
101110101 101110101
101111101 101111101
110000101 101
110001101 1101
110010101 110010101
110011101 11101
110100101 100101
110101101 110101101
110110101 110110101
110111101 110111101
111000101 101
111001101 111001101
111010101 111010101
111011101 111011101
111100101 111100101
111101101 111101101
111110101 111110101
111111101 111111101
1000000101 101
1000001101 1101
1000010101 10101
1000011101 11101
1000100101 100101
1000101101 101101
1000110101 110101
1000111101 111101
1001000101 101
1001001101 1001101
1001010101 1010101
1001011101 1011101
1001100101 1100101
1001101101 1101101
1001110101 1001110101
1001111101 1111101
1010000101 101
1010001101 1101
1010010101 10010101
1010011101 11101
1010100101 100101
1010101101 10101101
1010110101 1010110101
1010111101 10111101
1011000101 101
1011001101 11001101
1011010101 1011010101
1011011101 11011101
1011100101 11100101
1011101101 1011101101
1011110101 1011110101
1011111101 1011111101
1100000101 101
1100001101 1101
1100010101 10101
1100011101 11101
1100100101 100101
1100101101 101101
1100110101 1100110101
1100111101 111101
1101000101 101
1101001101 1001101
1101010101 1101010101
1101011101 1011101
1101100101 1100101
1101101101 1101101101
1101110101 1101110101
1101111101 1101111101
1110000101 101
1110001101 1101
1110010101 1110010101
1110011101 11101
1110100101 100101
1110101101 1110101101
1110110101 1110110101
1110111101 1110111101
1111000101 101
1111001101 1111001101
1111010101 1111010101
1111011101 1111011101
1111100101 1111100101
1111101101 1111101101
1111110101 1111110101
1111111101 1111111101
10000000101 101
10000001101 1101
10000010101 10101
10000011101 11101
10000100101 100101
10000101101 101101
10000110101 110101
10000111101 111101
10001000101 101
10001001101 1001101
10001010101 1010101
10001011101 1011101
10001100101 1100101
10001101101 1101101
10001110101 1110101
10001111101 1111101
10010000101 101
10010001101 1101
10010010101 10010101
10010011101 11101
10010100101 100101
10010101101 10101101
10010110101 10010110101
10010111101 10111101
10011000101 101
10011001101 11001101
10011010101 10011010101
10011011101 11011101
10011100101 11100101
10011101101 10011101101
10011110101 11110101
10011111101 10011111101
10100000101 101
10100001101 1101
10100010101 10101
10100011101 11101
10100100101 100101
10100101101 101101
10100110101 100110101
10100111101 111101
10101000101 101
10101001101 1001101
10101010101 101010101
10101011101 1011101
10101100101 1100101
10101101101 101101101
10101110101 10101110101
10101111101 101111101
10110000101 101
10110001101 1101
10110010101 110010101
10110011101 11101
10110100101 100101
10110101101 110101101
10110110101 10110110101
10110111101 110111101
10111000101 101
10111001101 111001101
10111010101 10111010101
10111011101 111011101
10111100101 111100101
10111101101 10111101101
10111110101 10111110101
10111111101 10111111101
11000000101 101
11000001101 1101
11000010101 10101
11000011101 11101
11000100101 100101
11000101101 101101
11000110101 110101
11000111101 111101
11001000101 101
11001001101 1001101
11001010101 1010101
11001011101 1011101
11001100101 1100101
11001101101 1101101
11001110101 11001110101
11001111101 1111101
11010000101 101
11010001101 1101
11010010101 10010101
11010011101 11101
11010100101 100101
11010101101 10101101
11010110101 11010110101
11010111101 10111101
11011000101 101
11011001101 11001101
11011010101 11011010101
11011011101 11011101
11011100101 11100101
11011101101 11011101101
11011110101 11011110101
11011111101 11011111101
11100000101 101
11100001101 1101
11100010101 10101
11100011101 11101
11100100101 100101
11100101101 101101
11100110101 11100110101
11100111101 111101
11101000101 101
11101001101 1001101
11101010101 11101010101
11101011101 1011101
11101100101 1100101
11101101101 11101101101
11101110101 11101110101
11101111101 11101111101
11110000101 101
11110001101 1101
11110010101 11110010101
11110011101 11101
11110100101 100101
11110101101 11110101101
11110110101 11110110101
11110111101 11110111101
11111000101 101
11111001101 11111001101
11111010101 11111010101
11111011101 11111011101
11111100101 11111100101
11111101101 11111101101
11111110101 11111110101
11111111101 11111111101
100000000101 101
100000001101 1101
100000010101 10101
100000011101 11101
100000100101 100101
100000101101 101101
100000110101 110101
100000111101 111101
100001000101 101
100001001101 1001101
100001010101 1010101
100001011101 1011101
100001100101 1100101
100001101101 1101101
100001110101 1110101
100001111101 1111101
100010000101 101
100010001101 1101
100010010101 10010101
100010011101 11101
100010100101 100101
100010101101 10101101
100010110101 100010110101
100010111101 10111101
100011000101 101
100011001101 11001101
100011010101 11010101
100011011101 11011101
100011100101 11100101
100011101101 100011101101
100011110101 11110101
100011111101 11111101
100100000101 101
100100001101 1101
100100010101 10101
100100011101 11101
100100100101 100101
100100101101 101101
100100110101 100110101
100100111101 111101
100101000101 101
100101001101 1001101
100101010101 101010101
100101011101 1011101
100101100101 1100101
100101101101 101101101
100101110101 101110101
100101111101 101111101
100110000101 101
100110001101 1101
100110010101 110010101
100110011101 11101
100110100101 100101
100110101101 110101101
100110110101 100110110101
100110111101 110111101
100111000101 101
100111001101 111001101
100111010101 100111010101
100111011101 111011101
100111100101 111100101
100111101101 100111101101
100111110101 111110101
100111111101 100111111101
101000000101 101
101000001101 1101
101000010101 10101
101000011101 11101
101000100101 100101
101000101101 101101
101000110101 110101
101000111101 111101
101001000101 101
101001001101 1001101
101001010101 1010101
101001011101 1011101
101001100101 1100101
101001101101 1101101
101001110101 1001110101
101001111101 1111101
101010000101 101
101010001101 1101
101010010101 10010101
101010011101 11101
101010100101 100101
101010101101 10101101
101010110101 101010110101
101010111101 10111101
101011000101 101
101011001101 11001101
101011010101 101011010101
101011011101 11011101
101011100101 11100101
101011101101 101011101101

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Written by meditationatae

September 16, 2017 at 8:17 pm

Posted in History

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