I am trying to solve Modified Fibonacci golden nuggets.
The generating function could be written as: $$A_G(x)=\frac{x(3x+1)}{1-x-x^2}$$
Let it be some $y\in\mathbb N$, then $$x(3x+1)=y(1-x-x^2)\\ x^2(3+y)+x(1+y)-y=0\\ x=\frac{-(1+y)+\sqrt{(1+y)^2-4(-y)(3+y)}}{2(3+y)}$$ So for $x\in\mathbb Q$, for some $k\in\mathbb N$ $$k^2=5y^2+14y+1\\ 5k^2=25y^2+70y+5=(5y+7)^2-44\\ X^2-5Y^2=44\text{ where }X=5y+7,\; Y=k$$ So I need to solve the diophantine equation $x^2-5y^2=44$. I note that $(7, 1)$ is a solution. Then $x^2-5y^2=1$ has a solution $(9, 4)$. And we know that other solutions will be given by $$x+y\sqrt5=\pm(7+\sqrt5)(9+4\sqrt5)^n$$ This gives me the list: $$\color{red}{(7,1)},(83,37),(767,665),(6923,9053),...$$ But this was not sufficient and I came to know we need to find other fundamental solutions which form their own classes like the one above for $(7, 1)$
Can someone illustrate how to find other solutions/fundamental solutions. Also any help regarding the problem would be great.
Solved! The sum is 5,673,835,352,990
The first 30 Golden nuggets are
$2, 5, 21, 42, 152, 296, 1050, 2037, 7205, 13970, 49392, 95760, 338546, 656357, 2320437, 4498746, 15904520, 30834872, 109011210, 211345365, 747173957, 1448582690, 5121206496, 9928733472, 35101271522, 68052551621, 240587694165, 466439127882, 1649012587640, 3197021343560$
I noted that x converged to $-\phi$ and $-\phi+1$ so I used several approximations of the continued fractions $-1;1;1;1;\ldots$ for $-\phi$ and $-0;1;1;1\ldots$ for the second.
Then I plugged those values in the generating function and got the 30 integers.