Original Sequence Puzzle
Scott Sauyet
This is an original sequence puzzle. If you don't like sequence puzzles,
please leave this screen immediately. :-)
What are the next few terms of the following (infinite) sequence?:
1,3,7,8,10,12,14,15,20,24,27,29,30,33,34,37,40,...
I think this is fairly difficult, so I have prepared two hints. E-mail
if you would like (a) the slightly helpful hint or (b) the much more
helpful hint. (Or I guess (c) the solution. :->)
__ ___
(_ c o t | Are the last three words of ssauyet@math
__) a u y e | this sentence "used or mentioned?" .wesleyan.edu
Scott Sauyet
I wrote:
> This is an original sequence puzzle. If you don't like sequence puzzles,
> please leave this screen immediately. :-)
>
> What are the next few terms of the following (infinite) sequence?:
>
> 1,3,7,8,10,12,14,15,20,24,27,29,30,33,34,37,40,...
There were only two people asking for hints and no posted
responses. :-(
I'm bummed that this generated no response. When I sent it to
the oracle, Chris Cole liked it well enough to forward it to
Neil Sloane, who in turn liked it well enough to place it in
his database. That was a real upper!
So if anyone cares, below is the solution.
Solution:
2 4 5 6 9 11 13 16 17 18 19 21,. . .
o n e, t h r e e, s e v e n, e i g h t, t e n, . . .
^ ^ ^ ^ ^ ^ ^ ^ ^
1 3 7 8 10 12 14 15 20, . . .
In other words, the sequence is self-descriptive in the following
sense: Starting with 1, it lists the position of the vowels among
the alphabetic characters formed by spelling out the members of
this sequence.
Actually, I'd like to find a way of stating that description similar
to what Hofstadter (in Godel, Escher, Bach) calls Quinification,
creating self-reference without a "this sentence", in the way that
"'Preceeded by its own quotation,' preceeded by its own quotation" is
self-descriptive. That is, I'd like to have a description of the
sequence that doesn't use "this sequence" in it.
Chris Cole also suggested the variation:
1 3 5 6 8 9 12 14, . . .
z e r o, t w o, f o u r, s e v e n, . . .
^ ^ ^ ^ ^ ^ ^ ^
0 2 4 7 10 11 13 15,. . .
which counts the consonants and starts indexing at 0.
There are obviously many variations on this theme. Matthew Daly listed
several in other languages. I like my finite French version:
2
u n
^
1
We could do a binary sequence in which vowels in the alphabetic sequence
generate 1's, consonants 0's:
o n e, z e r o, o n e, z e r o, o n e, z e r o, o n e, o n e, . . .
1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1, . . .
And I'm sure others could find many variations of the theme.