Given Fibonacci numbers $F_n$, we have the known,
$$(F_n F_{n+3})^2+(2F_{n+1}F_{n+2})^2 = (F_{2n+3})^2$$
as well as Lucas numbers $L_n$,
$$(L_n L_{n+3})^2+(2L_{n+1}L_{n+2})^2 = (L_{2n+2}+L_{2n+4})^2$$
and made me wonder if other sequences obey similar relationships.
Q: Are there similar formulas for Pythagorean triples using the Jacobsthal numbers,
$$J_n = 0, 1, 1, 3, 5, 11, 21, 43, 85, \dots$$
Jacobsthal-Lucas numbers,
$$K_n = 2, 1, 5, 7, 17, 31, 65, 127, \dots$$
$$P_n = 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, \dots$$
$$Q_n = 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, \dots$$
or other well-known sequences?
P.S. Of course, we $\color{red}{\text{disregard}}$ the obvious,
$$(J_m^2-J_n^2)^2+(2J_mJ_n)^2 = (J_m^2+J_n^2)^2$$