Abstract In the present work, a new sequence of quaternions related to Gaussian Bronze numbers is defined and studied. Binet’s formula, generating function certain properties identities are provided. Tridiagonal matrices considered determine general term this sequence.

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A fast and simple O(logn) iteration algorithm for individual Lucas numbers is given. This is faster than using Fibonacci based methods because of the structure of Lucas numbers. Using a √ 5 conversion factor gives a faster Fibonacci algorithm because the speed up proposed in [5] also directly applies. A fast simple recursive algorithm for individual Lucas numbers is given that is O(logn).