Abstract The article considers sequences with the golden constant, primarily the classical Fibonacci series. The well-known Cassini and Catalan identities provide particular relations between squares and products of terms of these sequences. In the present work, we propose a simple and visual method that allows one to systematically obtain all possible representations of the square of a number as the square of a smaller number plus the product of two other numbers from the same sequence, with the resulting quadruple forming a recurrent series. For sequences of the form 0,a,a0,a,a (multiples of the Fibonacci sequence), we formulate a rule that relates the sum of the square of the third term and pairwise shifted products to the square of the penultimate term. We also construct representations of the terms themselves (without squaring) as a sum of two products with variable step. This work continues the tradition of Cassini and Catalan, offering a constructive algorithm that is easy to verify and suitable for further generalizations.
Emma Helmdach (Sun,) studied this question.