[seqfan] Re: Looking for an interpretation

israel at math.ubc.ca israel at math.ubc.ca
Wed Dec 23 23:42:44 CET 2015


With a(n) = (2n^2 + 12n + 5 - 5 (-1)^n)/8 
and b(n) = a(n+1) - a(n) = (2n + 7 + 5 (-1)^n)/4
(note that these are integers)
we do have a(n)/b(n) + 1 = b(n+1).
Since [x(1),y(1)] = [a(1),b(1)] = [3,1], we will get x(n) = a(n) 
and y(n) = b(n) for n >= 1.

Somewhat more generally, if 
a(n) = A n^2 + B n + C + D (-1)^n
and b(n) = a(n+1) - a(n) = 2 A n + A + B - 2 D (-1)^n
we will have a(n)/b(n) + 1 = b(n+1) as long as 
A = 1/4 and C = B^2 - 4 D^2 - 1/16.

Cheers,
Robert 

On Dec 23 2015, rkg wrote:

>I believe that I missed some earlier messages about this.
>
>The sequence should start with 0 (=0*3) instead of 1.
>
>The polynomial  x^4 - 2x^3 + 2x - 1 has roots 1,1,1,-1
>so that the nth term is of shape
>
>         (An^2 + Bn + C)*1^n + D(-1)^n
>
>and in this case is
>
>         (1/8)[2n^2 + 12n + 5(1 - (-1)^n)]
>
>I have no combinatorial connexion.  How did it
>arise?     R.
>
>On Fri, 18 Dec 2015, Bruno Berselli wrote:
>
>> I confirm the recurrence.
>> Bruno 
>>
>>    Il Venerdì 18 Dicembre 2015 12:48, Ron Hardin <rhhardin at att.net> ha 
>> scritto:
>> 
>>
>> a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>5, 
>> apparently. rhhardin at mindspring.com rhhardin at att.net (either)
>> 
>>
>> 
>>       From: Peter Luschny <peter.luschny at gmail.com>
>> To: "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu> 
>> Sent: Friday, December 18, 2015 5:06 AM
>> Subject: [seqfan] Looking for an interpretation
>>   
>> Consider the algorithm:
>>
>> x, y = 1, 2
>> repeat:
>>   print x
>>   x, y = x + y, x//y + 1
>>
>> The assignment is simultaneous and x//y means floor(x/y).
>> This leads to:
>>
>> 1, 3, 4, 8, 10, 15, 18, 24, 28, 35, 40, 48, 54, 63, 70, ...
>>
>> Can you contribute a combinatorial interpretation of this sequence?
>>
>> Cheers, Peter
>>
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