To celebrate my birthday, I like to find interesting number theoretic properties of my new age. My upcoming 61st birthday was challenging, but then I noticed that $61 = 5^2 + 6^2 = 5^3 - 4^3$, the sum of two consecutive squares and the difference of two consecutive cubes. I wondered what other numbers had this property; that is, the integer solutions to $a^2+(a+1)^{2} = (b+1)^3 - b^3$ or equivalently $2a^2+2a = 3b^2+3b$. I ran an experiment in Matlab and got the following striking results.

$61 = 5^2 + 6^2 = 5^3 - 4^3$

$5941 = 54^2 + 55^2 = 45^3 - 44^3$

$582121 = 539^2 + 540^2 = 441^3 - 440^3$

$57041881 = 5340^2 + 5341^2 = 4361^3 - 4360^3$

$5589522181 = 52865^2 + 52866^2 = 43165^3 - 43164^3$

$547716131821 = 523314^2+523315^2 = 427285^3-427284^3$

$53670591396241 = 5180279^2+5180280^2 = 4229681^3 - 4229680^3$

It is easy to show that $a/b$ is close to $\sqrt{3/2}$ but not very close; the error is $O(1/b)$. The ratio between successive values of $a$ is monotonically decreasing; its last value is 9.898987988091280 so perhaps it is converging to 980/99.

Does anyone know what is going on here?