Sequence in context

[Rainer Rosenthal]

franktaw at netscape.net wrote:
> The "context" in "sequences in context" refers only to the 
> lexicographical ordering
> of the sequence.  Sequences are ordered (based on the absolute values) 
> from the
> first value greater than 1 (so 1,2,4,... comes after 2,3,5,...).  
> Sequences with only
> zeros and ones are ordered simply from the first value in the sequence 
> -- which
> means that all of them sort before any sequence containing a value 
> greater than
> one.

Oh, great! I never realised this interesting feature. Opens my eyes now,
thank you. When I submitted A136617 I found a false claim in a comment
for A083088, saying that this sequence had the properties of A136617.
This claim was removed by njas, but I thought it worth mentioning the
matching of the first 23 terms. Now I see A083088 as one of the sequences
"in context" and are astonished by the even better matching with e.g.
sequence A083089. I can't believe there is a deeper connection, but it's
nice.

No wonder I became interested in the next neighbour above, i.e. in A081223.
This finding alarmed me and I will have to find out more about the
relationship as it is close to the same theme 'Euler gamma'.

I remember the interesting connections to the continued fraction expansion
of  e  and the most interesting ideas of David W. Cantrell. Hopefully I will
return to this subject again.

Best regards, and special greetings to David,
Rainer Rosenthal




I am having problems verifying A135581 with a first discrepancy at a(16)=57.
This means I think that the 16th number with 25 divisors is 10556001 which
has the divisors 1, 3, 9, 19, 27, 57, 81, 171, 361, 513, 1083,... and the 5th
of these is 27, not 57. The 17th number with 25 divisors is 11316496 which
has the divisors 1, 2, 4, 8, 16, 29, 58, 116, 232, 464, 841, 1682... and the
5th of these is 16, not 29.

My version of A135581 is
6, 8, 8, 15, 21, 11, 13, 27, 16, 35, 16, 27, 16, 27, 55, 27, 16, 16, 16, 65, 27,
16, 77, 16, 85, 16, 29, 91, 31, 16, 95, 16, 37, 115, 16, 119, 16, 41, 43, 133,
16, 47, 16, 143, 125, 16, 125, 16, 53, 161, 16, 59, 16, 61, 125, 187, 16, 67,
16, 203, 125, 16, 209, 71, 16, 125, 217, 16, 73, 221, 16, 125, 79, 247, 81, 253,
16, 259, 16, 125, 81, 16, 16, 287, 81, 125, 16, 299, 301, 16, 81, 125, 81, 16,
319, 81, 323, 16, 81, 329, 16, 125, 81, 341, 16, 125, 16, 16, 125, 343, 377,
81, 16, 16, 391, 81, 16, 125, 16, 403, 407, 81, 343, 125, 81, 16, 343, 437
based on the following list of numbers with 25 divisors
1 1296
2 10000
3 38416
4 50625
5 194481
6 234256
7 456976
8 1185921
9 1336336
10 1500625
11 2085136
12 2313441
13 4477456
14 6765201
15 9150625
16 10556001
17 11316496
18 14776336
19 16777216
20 17850625
21 22667121
22 29986576
23 35153041
24 45212176
25 52200625
26 54700816
27 57289761
28 68574961
29 74805201
30 78074896
31 81450625
32 126247696
33 151807041
34 174900625
35 193877776
36 200533921
37 221533456
38 228886641
39 276922881
40 312900721
41 322417936
42 395254161
43 406586896
....

Can anyone verify/falsify these results?

Richard



Maximilian,

Thanks for your comments: Neil has been kind enough to add the sequence as
A136257.

(0) It is difficult to be precise about what is being counted in such chess
related sequences: by the number of possible plays on the nth move, I mean
the total number of legal lines of play for white under the rules of mirror
chess at a depth of n moves from the standard initial position.

(1) If white cannot play a legal move under the rules of mirror chess then I
consider the game to be a draw.

(2) I think that requiring white to play so that a draw can never happen
would require knowledge of all possible future lines of play in the game
tree!!

(3) Another interesting sequence is m(n) = number of mating plays on the nth
move in mirror chess. It is known that m(4) = 3 as explained here:
http://youtube.com/watch?v=C-D96Db4U2w&feature=related ("Mirror Mate in 4").

I am collecting mirror chess related links at
http://www.woomerang.com/mchess/.

Jeremy

On 18/4/08 14:41, "Maximilian Hasler" <maximilian.hasler at gmail.com> wrote:

> On Fri, Apr 18, 2008 at 8:42 AM, Jeremy Gardiner
> <jeremy.gardiner at btinternet.com> wrote:
>> consider the number of possible plays on the nth move in Mirror Chess in
>> which Black's play is always the mirror image of White (White must either
>> mate or play such that Black can mirror the move).
>> I find 20, 433, ...
> 
> some comments:
> 
> (0) first I was confused by "number of possible plays on the n-th
> move" because I didn't see how you choose the preceding moves. Now I
> understand that you speak of the "cumulated" number of moves.
> Also it appears you count twice the same move if it happens at the
> same depth but through different "history" of preceding moves - this
> convention is not necessarily the only one that makes sense.
> 
> (1) what happens if white cannot play such a move ?
> 
> (2) shouldn't white moves be required to be such that (1) cannot happen?
> (i.e. the game can come to a "normal" end) -
> but then the question of whether or not a move is allowed might become
> quite nontrivial.
> 
> (3) I suppose there is a minimal winning strategy for white (i.e.
> sequence of moves such that black will be mate in a minimal number of
> moves); in that case one could add the new sequence :
> number of possible moves (for the side to move) at the n-th half move.
> and idem for other extremal sequences (e.g. choose the move such that
> there is a minimal / maximal number of moves possible at the next
> move, ....)
> 
> Maximilian




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