[seqfan] Re: A090466

[Charles Greathouse]

I agree, (number of terms up to x)/x should converge. The limit could be
interpreted as the probability that a random number is a nontrivial
figurate number. A first approximation of the limit is

1 - product_{n >= 3} 1 - 2/(n^2 - n) = 2/3 = 0.6666...

and a better approximation is

1 - product_{n >= 3, n ≠ 6} 1 - 2/(n^2 - n) = 9/14 = 0.6428...

noting that all hexagonal numbers are also triangular.

Charles Greathouse
Case Western Reserve University

On Thu, Jan 19, 2017 at 8:48 AM, Daniel <kimpire at yahoo.com> wrote:

> Hi all,
>
> A090466 has the following fascinating tidbit:
>
> "Number of terms less than or equal to 10^n: 3, 57, 622, 6357, 63889,
> 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166,
> 640362343980, ..., . - Robert G. Wilson v, May 29 2014"
>
> I'm guessing based on looks alone (I'm not a mathematician) that this
> converges rather than diverges. If it does, does anybody know what it might
> be converging to, and what the significance of that fraction might be?
>
> -Daniel Sterman
>
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