[seqfan] Re: Pythagorean areas and congrua

[Bob Selcoe]

Hi Robert and Seqfans,

Perhaps just add the correct terms to A057102, and include the formula a(n)
= 4*A009112(n)??   I suppose if someone here posts on Wikipedia and
Mathworld, they could correct it there, too.

Also, interestingly, it looks like all of these terms can be generated by
sets of 3 "base" numbers and their multiples.

For example, {7,5,1} is:

7^2 - 5^2 = 5^2 - 1^2 = 24
14^2 - 10^2 = 10^ - 2^2 = 96
21^2 - 15^2 = 15^ - 3^2 = 216
28^2 - 20^2 = 20^2 - 4^2 = 384
etc.

Then {17,13,7}:

17^2 - 13^2 = 13^2 - 7^2 = 120
34^2 - 26^3 = 26^2 - 14^2 = 480
51^2 - 39^2 = 39^2 - 21^2 = 1080 (also not included in A057102)
etc.

Additional sets are {23,17,7}, {31,25,17}, {49,41,31}...

Best Wishes,
Bob Selcoe


--------------------------------------------------
From: <israel at math.ubc.ca>
Sent: Friday, April 03, 2015 7:19 PM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] Pythagorean areas and congrua

> A073120 has the Name "Areas of right triangles with integer sides". But it 
> doesn't contain the areas of all Pythagorean triangles, just those with 
> sides of the form (2mn, m^2-n^2, m^2+n^2). For example, the sequence does 
> not contain 54, the area of the Pythagorean triangle with sides (9,12,15). 
> The sequence with the areas of all Pythagorean triangles is A009112.
>
> A057102 has the Name "Congrua (possible solutions to the congruum 
> problem): numbers n such that there are integers x, y and z with n = 
> x^2-y^2 = z^2-x^2." Now n is a "congruum" iff n/4 is the area of a 
> Pythagorean triangle. So we should have A057102(n) = 4*A009112(n). For 
> example, 4*54 = 216 should be in the sequence: 216 = 15^2 - 3^2 = 21^2 - 
> 15^2. Unfortunately, it's not in the Data: the Formula for A057102 is a(n) 
> = 4 * A073120(n) and the Data seem to match that.
>
> The sequence 4*A009112(n) does not seem to be in the OEIS.  It should 
> start
> 24, 96, 120, 216, 240, 336, 384, 480, 600, 720, 840, 864, 960.
>
> This has propagated to Wikipedia: http://en.wikipedia.org/wiki/Congruum 
> says
>
> The first few congrua are:
>
>    24, 96, 120, 240, 336, 384, 480, 720, 840, 960 ¨ (sequence A057102 in 
> OEIS).
>
> and similarly MathWorld: http://mathworld.wolfram.com/Congruum.html
>
> So, what to do?
>
> Cheers,
> Robert Israel
>
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