# [seqfan] Re: A000134

Robert Israel israel at math.ubc.ca
Fri Aug 28 09:21:18 CEST 2009

```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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```