# [seqfan] Re: Formulas for two sequences, A060574 and A067498 ?

Андрей Заболотский zabolotis at mail.ru
Tue Mar 29 10:43:27 CEST 2022

```> Does any on SeqFan has this book at their disposal?

If I were to tell people to use copyright-violating websites, I would suggest typing the name of the book into the search bar at  https://libgen.is  . Since copyright violation is illegal in many countries, I of course won’t suggest that.

>Вторник, 29 марта 2022, 12:27 +04:00 от Antti Karttunen <antti.karttunen at gmail.com>:
>
>On 3/29/22, M. F. Hasler < oeis at hasler.fr > wrote:
>> Yes, for the tower of Hanoi there is a very simple algorithm that easily
>> translates to a formula :
>> on the first, third, fifth,... move, move the smallest disc to the next
>> peg,
>> so the smallest disc is on peg (ceil(n/2) mod 3) after the n-th move,
>> which yields departure and arrival peg number for any of these moves.
>> On the 2nd, 4th, 6th... move, make the only possible move with a disc other
>> than the smallest one.
>> You can check that for moves numbered
>> A108269 < https://oeis.org/A108269 > Numbers of the form (2*m - 1)*4^k where
>> m >= 1, k >= 1.this corresponds to the move (a -> a+1 (mod 3)) where a =
>> floor(n/2+1) mod 3 is the peg just "after" the one on which you just put
>> the smallest disc,
>> for the other even moves, with numbers A036554
>> < https://oeis.org/A036554 > (binary
>> representation ends in an odd number of zeros),
>> it's the "reverse" move: (a+1 (mod 3) -> a).
>>
>> For the "number of reflections" sequence, i don't understand the given
>> definition, i will have a look at the sequence.
>
>It would help if we had an access to the reference:
>
>M.O. van Deventer in The Mathemagician and the Pied Puzzler edited by
>E. Berlekamp and T. Rodgers, AKPeters Publishers, pp. 245-251.
>
>Does any on SeqFan has this book at their disposal?
>
>The mined LODA-assembly program:
>https://github.com/loda-lang/loda-programs/blob/main/oeis/067/A067498.asm
>
>refers to A340081(n) = gcd(n-1, A003958(n)), where A003958 is: If n =
>Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).
>
>When the LODA-assembly is converted to PARI (taking heed of differing
>starting offsets, for LODA-programs it is always 0), we get this:
>
>A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
>A067498maybe(n) = 1+(2*((((n-1)^2)+gcd(n-1,A003958(n)))\2));
>
>I.e., the tentative formula for A067498 would be
>   a(n) = 1 + clear_the_lsb_of[ ((n-1)^2) + gcd(n-1,A003958(n)) ],
>but with only seventeen known terms, this could well be a spurious match.
>
>BTW, here's a similarly mined program for the Tower of Hanoi related sequence:
>https://github.com/loda-lang/loda-programs/blob/main/oeis/060/A060574.asm
>involving a period 6 (repeat [0, 1, 1, 1, 1, 0]) sequence A131719.
>
>(And my thanks to all seqfans who have put effort to find a formula
>for A060574.)
>
>
>Best regards,
>
>Antti
>
>
>>
>> - Maximilian
>>
>> *(PARI/GP) < http://pari.math.u-bordeaux.fr/gp.html >*
>> *Hanoi(n,p=[[]|n<-[1..3]])={p[1]=[1..n]; vector(2^n-1, n,** my(m = if(
>> bittest(n,0), [n\2%3, (n\2+1)%3]*
>> *, my(a = (n\2+1)%3, b = (a+1)%3); if(** !p[b+1] ||
>> (#p[a+1]&&p[a+1][1]<p[b+1][1])**,* *[a,b], **[b,a])/*if*/*
>> *)/*if*/ )/*my*/;** p[m[2]+1] = concat( p[m[1]+1][1],
>> p[m[2]+1]); p[m[1]+1] = p[m[1]+1] [^1]; m**)}*
>>
>>
>> On Mon, Mar 28, 2022, 08:06 Antti Karttunen < antti.karttunen at gmail.com >
>> wrote:
>>
>>> Cheers,
>>>
>>> Does anybody know formulas for these two sequences?
>>>
>>>  https://oeis.org/A060574
>>> Tower of Hanoi: using the optimal way to move an even number of disks
>>> from peg 0 to peg 2 or an odd number from peg 0 to peg 1, a(n) is the
>>> smallest disk on peg 1 after n moves (or 0 if there are no disks on
>>> peg 1).
>>>
>>>  https://oeis.org/A067498
>>> Maximum number of reflections for a ray of light which reflects at n
>>> points (reflecting more than once at most or all points).
>>>
>>> These recently appeared among the newly mined LODA-programs (see
>>>  https://github.com/loda-lang/ ), but at least for the latter there are
>>> so few terms present that it is highly likely a case of strong law of
>>> small numbers.
>>>
>>>
>>> Best regards,
>>>
>>> Antti
>>>
>>> --
>>> Seqfan Mailing list -  http://list.seqfan.eu/
>>>
>>
>
>--
>Seqfan Mailing list -  http://list.seqfan.eu/

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