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Abstract
The study explores the Hamming Weight Sequence (HWS), a fundamental concept in number theory and computing that counts the number of '1's in the binary representation of a natural number. By delving into the properties, generation rules, and theoretical connections of the HWS, the research highlights its mathematical significance and practical utility. The investigation employs descriptive and expository methodologies, systematically analyzing existing literature to establish a robust theoretical framework. Key findings include the formulation of the HWS's recurrence relations and its connections to sequences such as Gould's, Ruler, and Trailing Zero Counting sequences. Moreover, the study underscores the HWS's role in solving Diophantine equations, demonstrating its applicability in mathematical problem-solving contexts. While focused primarily on the HWS, the research contributes to advancing knowledge in discrete mathematics and number theory, paving the way for further studies on its applications across diverse domains.
