It gives me great pleasure NOT to say...
"----- pls wait a few days before posting answers or hints."

_______________________

Where n (>0) is a (positive) integer....

Some forms are obviously squares: n^2, n^4, n^6, .........

Why don't you suggest some forms that are (at least a bit) puzzling.

e.g. -------
n^2 -1, n^2 +1, n^4 + n, n^6 - n, ............

n^3, n^5, n^7 - 1, ............

n(n+1), n(n+1)(n+2), n(n+1)(n+2)(n+3), ............

Some of these forms are NEVER squares,
and others are sometimes squares.

___________________________________
(Sum of 2 squares) Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2

We've known about Squares ( and Pythagorean triples )
for so long that.... When we learn something
simple and new about them, it's like....
_______ about a favorite grand-father.

On Thursday, July 21, 2022 at 1:26:55 PM UTC-4, henh...@gmail.com wrote:
> It gives me great pleasure NOT to say...
> "----- pls wait a few days before posting answers or hints."
>
>
> _______________________
>
> Where n (>0) is a (positive) integer....
>
> Some forms are obviously squares: n^2, n^4, n^6, .........
>
> Why don't you suggest some forms that are (at least a bit) puzzling.
>
> e.g. -------
> n^2 -1, n^2 +1, n^4 + n, n^6 - n, ............
>
> n^3, n^5, n^7 - 1, ............
>
> n(n+1), n(n+1)(n+2), n(n+1)(n+2)(n+3), ............
>
>
> Some of these forms are NEVER squares,
> and others are sometimes squares.
>
>
>
> ___________________________________
> (Sum of 2 squares) Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2
>
>
> We've known about Squares ( and Pythagorean triples )
> for so long that.... When we learn something
> simple and new about them, it's like....
> _______ about a favorite grand-father.

The "polygonal numbers" are a set of sequences, one for each integer n. They are defined as x*((n-2)x+(4-n))/2.

So for example, the 3-gonal (triangle) numbers are x(x+1)/2; the 4-gonal (square numbers) are x*x, and the 5- (pentagonal) numbers are x(3x-1)/2.

Some triangle numbers are square; which ones is an interesting, and classic, problem that has been connected with Ramanujan.

All square numbers are square.

I don't know the answer for the other -gons: I suspect it's similar to the triangles (an interesting pattern of which ones are square). I can say that any that have solutions will have similar series of solutions, but I don't know if there are any which don't.

On Friday, July 22, 2022 at 9:21:23 AM UTC-7, Jonathan Dushoff wrote:
> On Thursday, July 21, 2022 at 1:26:55 PM UTC-4, henh...@gmail.com wrote:
> > It gives me great pleasure NOT to say...
> > "----- pls wait a few days before posting answers or hints."
> >
> >
> > _______________________
> >
> > Where n (>0) is a (positive) integer....
> >
> > Some forms are obviously squares: n^2, n^4, n^6, .........
> >
> > Why don't you suggest some forms that are (at least a bit) puzzling.
> >
> > e.g. -------
> > n^2 -1, n^2 +1, n^4 + n, n^6 - n, ............
> >
> > n^3, n^5, n^7 - 1, ............
> >
> > n(n+1), n(n+1)(n+2), n(n+1)(n+2)(n+3), ............
> >
> >
> > Some of these forms are NEVER squares,
> > and others are sometimes squares.
> >
> >
> >
> > ___________________________________
> > (Sum of 2 squares) Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2
> >
> >
> > We've known about Squares ( and Pythagorean triples )
> > for so long that.... When we learn something
> > simple and new about them, it's like....
> > _______ about a favorite grand-father.
> The "polygonal numbers" are a set of sequences, one for each integer n. They are defined as x*((n-2)x+(4-n))/2.
>
> So for example, the 3-gonal (triangle) numbers are x(x+1)/2; the 4-gonal (square numbers) are x*x, and the 5- (pentagonal) numbers are x(3x-1)/2.
>
> Some triangle numbers are square; which ones is an interesting, and classic, problem that has been connected with Ramanujan.
>
> All square numbers are square.
>
> I don't know the answer for the other -gons: I suspect it's similar to the triangles (an interesting pattern of which ones are square). I can say that any that have solutions will have similar series of solutions, but I don't know if there are any which don't.

thank you !!!!

there must be arguments both for and against calling 0 , 1 as squares.
there must be arguments both for and against calling 0 , 1 as triangular numbers.

_____________________

James Tanton on Twitter:

The squangular numbers:
1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (numbers that are both square and triangular).

............... are both square [n^2] and triangular [n(n+1)/2].
Next ten? Is there a formula for them? http://jamestanton.com/?p=596

In article <9ad9c317-744e-4191-8be6-18749644fd9an@googlegroups.com>,
henh...@gmail.com <henhanna@gmail.com> wrote:

> The squangular numbers:
> 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (numbers that
>are both square and triangular).
>
>.............. are both square [n^2] and
>triangular [n(n+1)/2].
> Next ten? Is there a formula for them?

Strange but true:

(sinh(2*n*asinh(1)))**2 / 8

The Online Encyclopedia of Integer Sequences is your friend:

https://oeis.org/A001110

-- Richard

On Sat, 23 Jul 2022 17:48:03 +0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote:

>In article <9ad9c317-744e-4191-8be6-18749644fd9an@googlegroups.com>,
>henh...@gmail.com <henhanna@gmail.com> wrote:
>
>> The squangular numbers:
>> 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (numbers that
>>are both square and triangular).
>>
>>.............. are both square [n^2] and
>>triangular [n(n+1)/2].
>> Next ten? Is there a formula for them?
>
>Strange but true:
>
> (sinh(2*n*asinh(1)))**2 / 8
>
>The Online Encyclopedia of Integer Sequences is your friend:
>
> https://oeis.org/A001110
>
>-- Richard

I would say 'Incredible' if it were just proposed like that!

How that combination of sinh and arcsinh always
conspire to produce an integer is almost unbelievable.

Thanks for tracking this down. One for the books.

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