We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following identities? I am new to this site and hopefully, I will have good replies.

1) Sum of the terms:

$$H_1 + H_2 + H_3 + \ldots H_n = H_{n+2} – (m+3)$$

2) For alternate terms, we see the following:

$$H_1 + H_3 + H_5 + \ldots +H_{2n-1} = H_{2n} – 2$$ and

$$H_2 + H_4 + H_6 + \ldots + H_{2n} = H_{2n+1} – (m + 1)\;.$$

3) At the same time, if we square each terms, we see the:

$$H_1^2 + H_2^2 + H_3^2 + \ldots + H_n^2 = H_n H_{n+1} – 2 (m + 1)\;.$$

4) Finally, this interesting identity is valid:

$$H_{2n} = \sum_{k=0}^{n}\binom{n}{k}H_{n-k}$$