# [seqfan] Re: unlabeled Motzkin numbers

Max Alekseyev maxale at gmail.com
Fri Oct 29 06:21:39 CEST 2010

```First 50 unlabeled Motzkin numbers:

[1, 2, 2, 4, 5, 12, 19, 46, 95, 230, 528, 1320, 3219, 8172, 20714,
53478, 138635, 363486, 957858, 2543476, 6788019, 18218772, 49120019,
133036406, 361736109, 987316658, 2703991820, 7429445752, 20473889133,
56579632732, 156766505691, 435424359838, 1212195903863, 3382021652006,
9455173413478, 26485223356764, 74324776203271, 208938316077196,
588326898627101, 1659205016382730, 4686290685410777,
13254876473888474, 37541445026997948, 106465430270105760,
302303935789401521, 859396117355475964, 2445882966702428972,
6968676526031919574, 19875504907920560757, 56744175137426181858]

Also, here is the number aperiodic configurations (that are not
invariant w.r.t. rotation by any angle < 2*pi):

[1, 0, 1, 1, 4, 6, 18, 36, 92, 209, 527, 1269, 3218, 8063, 20701,
53209, 138634, 362789, 957857, 2541735, 6787960, 18214250, 49120018,
133024306, 361736098, 987284765, 2703991469, 7429359867, 20473889132,
56579399002, 156766505690, 435423724404, 1212195901528, 3382019908247,
9455173413411, 26485218540719, 74324776203270, 208938302751504,
588326898610718, 1659204979313884, 4686290685410776,
13254876370356655, 37541445026997947, 106465429980232687,
302303935789281990, 859396116541094125, 2445882966702428971,
6968676523737317241, 19875504907920560700, 56744175130946379709]

Max

On Thu, Oct 28, 2010 at 11:57 PM, Max Alekseyev <maxale at gmail.com> wrote:
> More terms:
>
> 1, 2, 2, 4, 5, 12, 19, 46, 95, 230, 528
>
> I'd be grateful if somebody double check these values.
>
> Regards,
> Max
>
> On Thu, Oct 28, 2010 at 10:16 PM, Max Alekseyev <maxale at gmail.com> wrote:
>> Manually, I've got that the sequence of unlabeled Motzkin numbers starts with:
>> 1,2,2,4,5,12,19,...
>> and this sequence is not in the OEIS.
>>
>> In particular:
>> for n=1, there is one configuration (the empty one) of period 1
>> for n=2, there are two configurations both of period 1
>> for n=3, there are one configuration of period 1 and one of period 3
>> for n=4, there are one configuration of period 1, two of period 2, and
>> one of period 4
>> for n=5, there are one configuration of period 1 and four of period 5
>> for n=6, there are one configuration of period 1, one of period 2,
>> four of period 3, and six of period 6
>> ( check: 1*1 + 1*2 + 4*3 + 6*6 = 51 = A001006(6) )
>>
>> For odd prime p, we have
>> a(p) = (A001006(p) - 1)/p + 1
>>
>> Max
>>
>> On Wed, Oct 27, 2010 at 2:40 PM, Max Alekseyev <maxale at gmail.com> wrote:
>>> Just out of curiosity, what would be an "unlabeled" version of Motzkin
>>> numbers A001006?
>>> That is, number of ways of drawing any number of nonintersecting
>>> chords joining n unlabeled equally spaced points on a circle (in other
>>> words, the points and chords are distinguished up to rotations of the
>>> circle).
>>> Is this sequence in the OEIS?
>>>
>>> Furthermore, is there a variant of counting up to rotations and
>>> reflections of the circle?
>>>
>>> Regards,
>>> Max
>>>
>>
>

```