2
votes
1answer
53 views

Find the generating function for a series , given a recurrence relation

I am solving a problem on an Online Judge. The problems solution boils down to find the solutions to the following recurrence relation: ...
6
votes
0answers
155 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
2
votes
3answers
115 views

Proof that $x^2 - 2y^2 = -1$ has a recurring solution for $x$

I read here about the following variation on Pell's equation: $$ x^2 - 2y^2 = -1.$$ According to Dario Alpern's solver, the equation has infinite integer ...
1
vote
1answer
120 views

Alternative solutions to $n^2+n = k^2+k + 2kn$

Consider this equation: $n^2+n = k^2+k + 2kn$ I want to find the set of non-negative integer n,k that satisfies the equation. I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
2
votes
2answers
85 views

Twice a triangle is triangle

The question is to prove that there are infinitely many triangular numbers $T_n$ where $2 \times T_n$ is also a triangular number, and give the first few as an example. My attempt: $$2 \cdot {x(x+1) ...
12
votes
3answers
279 views

How to find a “better description” (e.g. recurrence relation) for this sequence?

My solution to a problem in Project Euler required to solve this subproblem: find values of $k\in\mathrm{N}$ such that $3k^2+4$ is a perfect square. As I was writing a computer program, I just tried ...
1
vote
1answer
193 views

Recursive solution to a Diophantine equation

I'd like to find a recursive formula giving positive integer solutions to this Diophantine equation $$5L^2 - a^2 - 1 =0$$ It can be seen that I need $5L^2 - 1$ to be a square of a number $\in \mathbb ...
0
votes
0answers
482 views

Using recurrences to solve $3a^2=2b^2+1$

Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, ...
3
votes
2answers
213 views

Recurrence relation for $a_{n}$ where $6a_{n}$ and $10a_{n}$ are both triangular

According A180926, the elements of the set {$a:\exists m,n|60a=5n^2+5n=3m^2+3m$} satisfy the following recurrence relation: $$a_{n}=\frac{62a_{n-1}+1+\sqrt{(48a_{n-1}+1)(80a_{n-1}+1)}}{2}$$ ...