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

Jeffrey Shallit shallit at uwaterloo.ca
Tue Mar 29 01:49:25 CEST 2022

```For A060574 it is almost certainly 2-regular and the defining relations are

A060574 = a(n)

b(n) = n mod 3
c(n) = (n+1) mod 3
d(n) = (n+2) mod 3

a(n) = b(n)/3 - 2c(n)/3 + d(n)/3 + a(4n) - a(4n+1) + a(4n+2)

a(8n+2) = b(n) + c(n)

a(8n+3) = 2b(n)/3 + 5c(n)/3  - d(n)/3 + a(8n) - a(8n+1)

a(8n+5) = b(n)/3 + c(n)/3 + 4d(n)/3

a(8n+6) = -b(n)/3 - c(n)/3 + 2d(n)/3 + a(8n) + a(8n+1)

a(16n) = -4b(n)/3 - c(n)/3 - d(n)/3 - a(n) + 2a(4n) + a(4n+2)

a(16n+1) = 2b(n)/9 + 8c(n)/9 - 4d(n)/9 + a(4n+1)

a(16n+4) = 11b(n)/9 - c(n)/9 + 14d(n)/9

a(16n+8) = 10b(n)/9 + 4c(n)/9  - 2d(n)/9 + a(4n+2)

which allows computing a(n) from starting values.

However, I did not prove this rigorously and it's possible I made a
transcription error.

On 2022-03-28 5:55 AM, Antti Karttunen 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/

```