[seqfan] Post's tag system, continued
Neil Sloane
njasloane at gmail.com
Tue Aug 1 18:44:35 CEST 2017
The basic sequence, A284116, specifies that only words w that have a
FINITE orbit are to be considered (although normally an OEIS entry
would have an escape clause like "or -1 if there is a word of length n
with an infinite orbit").
But the present version is perfectly fine as long as there is a
comment saying how far people have searched for an infinite orbit.
Don, if there were an infinite orbit starting at length <= 46, would
your program have found it?
Assuming the answer is Yes, then we know
that A289670(n) + A289671(n) = 2^n (because for n <= 46, every word
either dies or goes into a cycle). Assuming that is true, I've
extended A289671 from your b-file for A289671.
Second question: If we number the bits in the word w from left to
right starting at bit 0, then as Don pointed out, the bits
1,2,4,5,7,8,.,,, (not a multiple of 3) are "junk DNA" and have no
affect on the outcome of w under the Post tag system.
This means that we can divide the counts in A2676570 and A289671 by 2
to the power f1(n), where f1(n) is the number of non-multiples of 3 in
the range 0 to n-1. That is, f1(n) = 2*floor((n-1)/3)+((n+2) mod 3).
Don, do you agree?
The reduced counts for terminating / cycling are now A289676 and A289677:
> r1:=[seq(b1[n]/2^f1(n),n=1..nops(b1))];
r1 := [2, 1, 1, 2, 2, 1, 4, 4, 3, 5, 4, 3, 10, 13, 12, 21, 18, 20, 43,
40, 39, 85, 71, 64, 146, 132, 116, 250, 231, 210, 462, 459, 438, 960,
990, 966, 2069, 2114, 2089, 4296, 4237, 4155, 8486, 8235, 8033, 16497]
> r2:=[seq(b2[n]/2^f1(n),n=1..nops(b1))];
r2 := [0, 1, 1, 2, 2, 3, 4, 4, 5, 11, 12, 13, 22, 19, 20, 43, 46, 44,
85, 88, 89, 171, 185, 192, 366, 380, 396, 774, 793, 814, 1586, 1589,
1610, 3136, 3106, 3130, 6123, 6078, 6103, 12088, 12147, 12229, 24282,
24533, 24735, 49039]
It would have been a miracle if either sequence had already been in
the OEIS, but no such luck!
The reduced counts are smaller than the originals of course , and make
it easier to see the growth (in clumps of 3). The six trisections of
these two sequences are:
> DECIMATE(r1,3,0);
[2, 2, 4, 5, 10, 21, 43, 85, 146, 250, 462, 960, 2069, 4296, 8486, 16497]
> DECIMATE(r1,3,1);
[1, 2, 4, 4, 13, 18, 40, 71, 132, 231, 459, 990, 2114, 4237, 8235]
> DECIMATE(r1,3,2);
[1, 1, 3, 3, 12, 20, 39, 64, 116, 210, 438, 966, 2089, 4155, 8033]
> DECIMATE(r2,3,0);
[0, 2, 4, 11, 22, 43, 85, 171, 366, 774, 1586, 3136, 6123, 12088, 24282, 49039]
> DECIMATE(r2,3,1);
[1, 2, 4, 12, 19, 46, 88, 185, 380, 793, 1589, 3106, 6078, 12147, 24533]
> DECIMATE(r2,3,2);
[1, 3, 5, 13, 20, 44, 89, 192, 396, 814, 1610, 3130, 6103, 12229, 24735]
They aren't in the OEIS (but I didn't run them through Superseeker).
Can anyone see any patterns?
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