[seqfan] Re: A000134

[David Wilson]

I would conjecture that there are other (very sparse) exceptions. If we 
knew the exceptions very far out, it would be an easy compute:

a(n) =
	round(pi*(n-1/4))+1 if n is an exception
	round(pi*(n-1/4) otherwise.

Robert Israel wrote:
> Maple confirms this, and will give you as many additional terms as you 
> wish:
> 
>> B:= BesselJ(0,n-1/2)*BesselJ(0,n+1/2);
> 
>> map(combine, asympt(B,n,4));
> 
> (cos(1)+sin(2*n))/Pi/n-1/4*cos(2*n)/Pi/n^2+1/32*(-sin(2*n)-4*sin(1))/Pi/n^3+O(1/(n^4))
> 
> Cheers,
> Robert Israel
> 
> On Thu, 27 Aug 2009, Gerald McGarvey wrote:
> 
>> This inspired me to look at the shape of the Bessel function of order 0,
>> in PARI:    b(n) = besselj(0,n-1/2)*besselj(0,n+1/2)
>>
>> The function b(n) appears to asymptotically approach the following function:
>>
>> f(n) = 1/(n*Pi) * (sin(2*n) + cos(1))
>>
>> The difference between these functions also looks like a damped sinusoid,
>> around   1/(4*n^2*Pi)*cos(2*n)
>>
>> Regards,
>> Gerald McGarvey
>>
>> At 10:03 PM 8/26/2009, David Wilson wrote:
>>> For most n, A000134(n) = round(pi*(n-1/4)). For 1 <= n <= 1000, the only
>>> exception is n = 2.
>>>
>>>
>>>
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