[math-fun] Re: A016142

[Richard Guy]

Sorry about that.  Can't now trace how
I made the mistake.  Evidently

0, 1, 12, 101, 760, 5481, 38839, ...

is a new sequence.  Formula

       (2*7^n -3*3^4 + 1)/6

Characteristic polynomial (x-7)(x-3)(x-1)

Manifestation: number of incongruent
integer-edged Heron triangles whose
circumdiameter is the product of  n
distinct primes of shape  4k + 1.

I believe that my remarks about the
number of such triangles that are
not right-angled are still correct,
viz.

(2*7^n - 6*3^n + 4)/6

Same recurrence.

0, 0, 8, 88, 720, 5360, 38488, 272328, ...

Eight times A016212.  Not in OEIS per se.

The number of nondegenerate right-angled
such triangles is A003462, tho that fact
is not noted there.

Originators:  Alex Fink & Richard Guy

Will someone do the necessary?  Thanks!

On Wed, 17 Aug 2005, David Wilson wrote:

> ----- Original Message ----- From: "Richard Guy" 
> <rkg at cpsc.ucalgary.ca>
> To: "math-fun" <math-fun at mailman.xmission.com>
> Cc: "Alex Fink" <finka at math.ucalgary.ca>; 
> <seqfan at ext.jussieu.fr>
> Sent: Monday, August 15, 2005 2:51 PM
> Subject: Re: [math-fun] Re: A016142
>
>
>> Alex Fink & I are now able to give the
>> right answers.  The number of incongruent
>> integer-edged Heron triangles whose
>> circumdiameter is the product of  n
>> distinct primes each of shape  4k + 1
>> is
>>       (2*7^n - 3*3^n + 1)/6
>> 
>> This is A016161 in OEIS.
>
> Not true.  Compute the sequence.
>
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