On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers

In the present paper, we obtain identities for Narayana numbers, like the sum of terms with even and odd subscripts, the sum of products of consecutive terms and the sum of squares of terms. Then, we find images $DN$ and $D^2N$ of $n$-tuple $N=(N_1, \ N_2,\ N_3, \ ... \ ,\ N_n)$ of Narayana numbers under a map $D:\mathbb{C} \rightarrow \mathbb{C}$ defined as $D(z_1, \ z_2, \ ... \ , \ z_n)=(\lvert z_2-z_1\rvert, \ \lvert z_3-z_2\rvert, \ ... \ ,\ \lvert z_n-z_{n-1}\rvert, \ \lvert z_n-z_1\rvert ).$ We are then determined the circulant, skew-circulant, and semi-circulant matrices of these images. We have been discovered Frobenius norms of these circulant matrices and relations among these norms. In addition, we find $DG$ and $D^2G$ by taking the $n$-tuple $G=(GN_1,\ GN_2, \ ... \ , \ GN_n)$ of Gaussian Narayana numbers. After that, we create circulant, semi-circulant, and skew-circulant matrices of $G, \ DG, \ D^2G$, determine their Frobenius norms, and derive relationships between them. Then, we obtain relations between norms of matrices of Narayana numbers and Gaussian Narayana numbers. Finally, coding and decoding methods with the use of circulant matrices of Narayana numbers and Gaussian Narayana numbers have been introduced.