Maximilian Hasler maximilian.hasler at gmail.com
Mon Oct 5 19:00:26 CEST 2009

```Another basic sequence related to this is
a(n)=n+(least square > n)
since this is the map you apply.

? vector(99,i,i+(sqrtint(i)+1)^2)

[5, 6, 7, 13, 14, 15, 16, 17, 25, 26, 27, 28, 29, 30, 31, 41, 42, 43,
44, 45, 46, 47, 48, 49, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 113, 114, 115,
116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 145, 146,
147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160,
161, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199]

%e a(4)=13 = 4+9 because 9 is the least square > 4.
(a(0)=1 could be prefixed.)

Another seq. not in the OEIS:
a(n)=n+(least square >= n)

? vector(99,i,i+(sqrtint(i-1)+1)^2)

[2, 6, 7, 8, 14, 15, 16, 17, 18, 26, 27, 28, 29, 30, 31, 32, 42, 43,
44, 45, 46, 47, 48, 49, 50, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 114, 115, 116,
117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 146, 147,
148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161,
162, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194,
195, 196, 197, 198, 199]

%e a(4)= 8 = 4+4 because 4 is the least square >= 4.
%e a(5)= 14 = 5+9 because 9 is the least square >= 5.

Maximilian

On Mon, Oct 5, 2009 at 7:47 AM,  <c.zizka at email.cz> wrote:
> Dear seqfans,
>
> for a given mapping :
> a(n) = a(n-1) + [smallest square > a(n-1)]     if a(n-1) is not divisible 2 ,
> else
> a(n) = a(n-1)/2
>
> Not proven, just trial -  there are 2 periodic orbits L1= {1--5--14--7--16--8--4--2--(1) } and L2 = {11--27--63--127--271--560--280--140--70--35--71--152--76--38--19--44--22--(11)} , their length is  L1=8 and L2=17, the number of steps needed to come to one of  them from some a(0) is fluctuating.
>
> Not sure what is the number and length of periodic orbits if the divisor in the mapping is not 2 , but some other positive integer.
> Does every such mapping converge to some periodic orbit ? (Seems yes)
>
> The basic sequence related to the example above is :
> a(n) = a(n-1) + [smallest square > a(n-1)]
> {1,5,14,30,66,147,316,640,1316,2685,5389,10865,...}. Does it make any sense to put this seq. into OEIS ?
>
> Ctibor
>
>
>
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>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```