in doubt of A135393 (circular permutations with constrained differences)

Fri Jan 11 11:42:09 CET 2008

sequence currently in the OEIS?
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Date: Fri, 11 Jan 2008 09:28:44 -0500
From: "Alexander Povolotsky" <apovolot at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: known (coincidental ?) artifact ?
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Hi,

I noticed that for quite a number consecutive integers starting from some
integer n greater than 6,
below described reveals some persisting pattern.

 Consider calculating (in decimal system):

floor(n*n/10) + 10

and then looking between the  difference of calculated above and "doubled"
value of n :

(floor(n*n/10) + 10)  - 2*n

and then look at the differences between in above expression for two
consecutive integers n and (n-1)

a(n) = ((floor(an*n/10) + 10) - 2*n) - ((floor((n-1)*(n-1)/10) + 10 -
2*(n-1))

(09:04) gp > a(n)=(floor(n*n/10) +10 -2*n) - (floor((n-1)*(n-1)/10) + 10
-2*(n-1))

(09:14) gp > a(0)
%27 = -2
(09:15) gp > a(1)
%28 = -2
(09:15) gp > a(2)
%29 = -2
(09:15) gp > a(3)
%30 = -2
(09:15) gp > a(4)
%31 = -1
(09:15) gp > a(5)
%32 = -1
(09:06) gp > a(6)
%1 = -1
(09:07) gp > a(7)
%2 = -1
(09:07) gp > a(8)
%3 = 0
(09:08) gp > a(9)
%4 = 0
(09:08) gp > a(10)
%5 = 0
(09:08) gp > a(11)
%6 = 0
(09:08) gp > a(12)
%7 = 0
(09:08) gp > a(13)
%8 = 0
(09:08) gp > a(14)
%9 = 1
(09:09) gp > a(15)
%10 = 1
(09:09) gp > a(16)
%11 = 1
(09:11) gp > a(17)
%12 = 1
(09:11) gp >
(09:11) gp > a(18)
%13 = 2
(09:11) gp > a(19)
%14 = 2
(09:12) gp > a(20)
%15 = 2
(09:12) gp > a(20)
%16 = 2
(09:12) gp > a(21)
%17 = 2
(09:12) gp > a(22)
%18 = 2
(09:12) gp > a(23)
%19 = 2
(09:13) gp > a(24)
%20 = 3
(09:13) gp > a(25)
%21 = 3
(09:13) gp > a(26)
%22 = 3
(09:13) gp > a(27)
%23 = 3
(09:13) gp > a(28)
%24 = 4

....

So you could notice that starting from a(4) and on - the difference is kept
the same for first 4 terms and then the difference gets  incremented by 1
and as such is kept constant for next 6 terms, then it increments by 1 again
and is kept the same for 4 terms, etc, etc ....

I am not sure for how long this pattern of two consecutively interchanging
"periods" of 4 and 6 is maintained but it lasts for quite a while ....

Is this well known and/or obvious ?

Thanks,
Alexander R. Povolotsky

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<div>Hi,</div>
<div> </div>
<div>I noticed that for quite a number consecutive integers starting from some integer n greater than 6, </div>
<div>below described reveals some persisting pattern.</div>
<div> </div>
<div> Consider calculating (in decimal system): </div>
<div> </div>
<div>floor(n*n/10) + 10 </div>
<div> </div>
<div>and then looking between the  difference of calculated above and "doubled" value of n :</div>
<div> </div>
<div>(floor(n*n/10) + 10)  - 2*n</div>
<div> </div>
<div>and then look at the differences between in above expression for two consecutive integers n and (n-1)</div>
<div> </div>
<div>a(n) = ((floor(an*n/10) + 10) - 2*n) - ((floor((n-1)*(n-1)/10) + 10 - 2*(n-1))</div>
<div> </div>
<div>(09:04) gp > a(n)=(floor(n*n/10) +10 -2*n) - (floor((n-1)*(n-1)/10) + 10 -2*(n-1))</div>
<div> </div>
<div>(09:14) gp > a(0)<br>%27 = -2<br>(09:15) gp > a(1)<br>%28 = -2<br>(09:15) gp > a(2)<br>%29 = -2<br>(09:15) gp > a(3)<br>%30 = -2<br>(09:15) gp > a(4)<br>%31 = -1<br>(09:15) gp > a(5)<br>%32 = -1<br>
(09:06) gp > a(6)<br>%1 = -1<br>(09:07) gp > a(7)<br>%2 = -1<br>(09:07) gp > a(8)<br>%3 = 0<br>(09:08) gp > a(9)<br>%4 = 0<br>(09:08) gp > a(10)<br>%5 = 0<br>(09:08) gp > a(11)<br>%6 = 0<br>(09:08) gp > a(12)
<br>%7 = 0<br>(09:08) gp > a(13)<br>%8 = 0<br>(09:08) gp > a(14)<br>%9 = 1<br>(09:09) gp > a(15)<br>%10 = 1<br>(09:09) gp > a(16)<br>%11 = 1<br>(09:11) gp > a(17)<br>%12 = 1<br>(09:11) gp ><br>(09:11) gp > a(18)
<br>%13 = 2<br>(09:11) gp > a(19)<br>%14 = 2<br>(09:12) gp > a(20)<br>%15 = 2<br>(09:12) gp > a(20)<br>%16 = 2<br>(09:12) gp > a(21)<br>%17 = 2<br>(09:12) gp > a(22)<br>%18 = 2<br>(09:12) gp > a(23)<br>%19 = 2
<br>(09:13) gp > a(24)<br>%20 = 3<br>(09:13) gp > a(25)<br>%21 = 3<br>(09:13) gp > a(26)<br>%22 = 3<br>(09:13) gp > a(27)<br>%23 = 3<br>(09:13) gp > a(28)<br>%24 = 4</div>
<div> </div>
<div>....</div>
<div> </div>
<div>So you could notice that starting from a(4) and on - the difference is kept the same for first 4 terms and then the difference gets  incremented by 1 and as such is kept constant for next 6 terms, then it increments by 1 again and is kept the same for 4 terms, etc, etc ....
</div>
<div> </div>
<div>I am not sure for how long this pattern of two consecutively interchanging "periods" of 4 and 6 is maintained but it lasts for quite a while ....</div>
<div> </div>
<div>Is this well known and/or obvious ?</div>
<div> </div>
<div>Thanks,</div>
<div>Alexander R. Povolotsky</div>

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