[seqfan] Re: 2-3 sequence puzzle

[Vladimir Reshetnikov]

Actually, there is a nice proof that this algorithm is indeed correct:
http://math.stackexchange.com/q/1196327/19661

--
Thanks
Vladimir

On Wed, Mar 18, 2015 at 1:34 PM, Vladimir Reshetnikov <
v.reshetnikov at gmail.com> wrote:

> Yes, you are right!
>
> I'm trying to write a program that can generate n-th term of this
> sequence. Here is my attempt (in Mathematica syntax):
>
> ClearAll[prepend, a];
> prepend[d_, n_] := d * 10^Ceiling[Log10[n]] + n;
>
> a[1] = 2; a[2] = 3; a[3] = 22; a[4] = 23; a[5] = 32; a[6] = 222;
> a[7] = 33; a[8] = 322; a[9] = 223; a[10] = 232; a[11] = 323; a[12] = 332;
>
> a[n_ /; OddQ[n]] := a[n] = prepend[2, a[(n - 1)/2]];
> a[n_ /; EvenQ[n]] := a[n] = prepend[3, a[(n - 2)/2]];
>
> Array[a, 100]
>
>
> I do not yet have a rigorous proof that this program is correct. Can
> anybody suggest such a proof?
>
> --
> Thanks
> Vladimir
>
>
> On Wed, Mar 18, 2015 at 7:49 AM, Bob Selcoe <rselcoe at entouchonline.net>
> wrote:
>
>> Hi Vladimir, Reinhard and Seqfans,
>>
>> More specifically than simply a permutation of 2-3 numbers, isn't this
>> the sequence of power towers in ascending order using all permutations of
>> 2s and 3s?
>>
>> 2
>> 3
>> 2^2 = 4
>> 2^3 = 8
>> 3^2 = 9
>> 2^(2^2) = 16
>> 3^3= 27
>> 3^(2^2) = 81
>> 2^(2^3) = 256
>> 2^(3^2) = 512
>> ...
>>
>> This would make a nice OEIS entry (it's not already there!).
>>
>> Best,
>> Bob Selcoe
>>
>> --------------------------------------------------
>> From: "Reinhard Zumkeller" <reinhard.zumkeller at gmail.com>
>> Sent: Wednesday, March 18, 2015 6:11 AM
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Cc: <seqfan at seqfan.eu>
>> Subject: [seqfan] Re: 2-3 sequence puzzle
>>
>>
>>  Dear Vladimir,
>>>
>>> thanks for this nice challenge!
>>>
>>> My solution: it's a permutation of 2-3 numbers based on your
>>> https://oeis.org/A185969
>>>
>>> Please have a look at draft https://oeis.org/draft/A248907 and add some
>>> substance and a more explicit definition; the provisional definition
>>> should
>>> be just a comment.
>>>
>>> Best regards
>>> Reinhard
>>>
>>>
>>> 2015-03-18 0:11 GMT+01:00 Vladimir Reshetnikov <v.reshetnikov at gmail.com
>>> >:
>>>
>>>  Dear Seqfans,
>>>>
>>>> Here is a sequence consisting of all non-empty finite strings of digits
>>>> 2
>>>> and 3 without any duplicates. It begins as:
>>>>
>>>> 2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332, 2222, 3222, 233,
>>>> 333,
>>>> 2322, 3322, 2223, 3223, 2232, 3232, 2323, 3323, 2332, 3332, 22222,
>>>> 32222,
>>>> 23222, 33222, 2233, 3233, 2333, 3333, 22322, 32322, 23322, 33322, 22223,
>>>> 32223, 23223, 33223, 22232, 32232, 23232, 33232, 22323, 32323, 23323,
>>>> 33323, 22332, 32332, 23332, 33332, 222222, 322222, 232222, 332222,
>>>> 223222,
>>>> 323222, 233222, 333222, 22233, 32233, 23233, 33233, 22333, 32333, 23333,
>>>> 33333, 222322, ...
>>>>
>>>> Can you guess the order in what they appear?
>>>>
>>>> --
>>>> Vladimir
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>>
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>>>
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>>>
>>>
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>>
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>
>