Lucas Representations and Zeckendorf Structures of Constants in the Ramanujan and Chudnovsky 1/π Series

We examine the standard Zeckendorf expansions, with respect to the Lucas numbers, of the integer constants appearing in the classical 1/π series of Ramanujan and Chudnovsky. Our work yields four key, rigorously verified results: We prove a new identity for the sum of every third Lucas number, which provides a precise theoretical explanation for the structure of the Zeckendorf decomposition of the constant 1103. We show that the indices in the decomposition of 1103 form an arithmetic progression with a common difference of −3. We provide complete and verified standard Zeckendorf decompositions for all other constants in both the Ramanujan and Chudnovsky series. We identify a length-4 arithmetic progression within the decomposition of the constant 545140134, and explicitly distinguish these canonical decompositions from simpler algebraic identities, such as 128156 = 2(L₂₃ − 1). This paper is presented as a self-contained collection of observations, free from any external geometric or conjectural framework.