# [seqfan] Re: Problem submitting a sequence

Max Alekseyev maxale at gmail.com
Sat Nov 5 17:55:28 CET 2011

```Ops. I've got somehow wrong coefficient 4 instead of 8 for triangular numbers.
Let me start once again:

If a number m is triangular then 8m+1 is a square.
If a number m is decagonal then 16m+9 is a square.

Hence we have a system of equations:
8m+1 = x^2
16m+9 = y^2
which implies
y^2 - 2x^2 = 7

The solutions to this equation are given by x in A077442. Solutions m
are obtained from the terms of A077442 by the formula:
m = (A077442(n)^2 - 1) / 8.
However not every term belongs to decagonal numbers. In addition, we
want sqrt(16*m+9)=sqrt(2*A077442(n)^2+7) be congruent to 5 modulo 16.

The sequence of numbers that are both triangular and decagonal starts with:
1, 10, 1540, 11935, 1777555, 13773376, 2051297326, 15894464365, ...

Regards,
Max

On Sat, Nov 5, 2011 at 8:40 PM, Max Alekseyev <maxale at gmail.com> wrote:
> On Fri, Nov 4, 2011 at 6:31 PM, Alonso Del Arte
> <alonso.delarte at gmail.com> wrote:
>> Would that be numbers that are both triangular and decagonal? Al
>
> If a number m is triangular then 4m+1 is a square.
> If a number m is decagonal then 16m+9 is a square.
>
> Hence we have a system of equations:
> 4m+1 = x^2
> 16m+9 = y^2
> which implies
> y^2 - 4x^2 = 5
> that is
> (y-2x)(y+2x)=5
> implying that the only solution in nonnegative x,y is (x,y)=(1,3).
>
> Therefore, there is the only number that is both triangular and
> decagonal which is 0.
>
> Regards,
> Max
>

```