The diamond lattice

This comprehensive pedagogical guide provides an in-depth exploration of the Lichten-berg sequence (OEIS A001045) with extensive teaching support for independent learners. Every section includes detailed teaching boxes with prerequisites and learning objectives, numerous worked examples with step-by-step calculations, intuition boxes explaining the mathematics, extensive practice problems with complete solutions, warnings about common errors, historical context, and real-world applications. We use ONLY the mathematically correct sum rule definition a(n) = 2n−1 a(n 1) and explicitly document incorrect formula-tions found in some literature. The guide proves key properties including ratio convergence to 2 with exponentially decaying error ( 3/4 n ), presents the novel Diamond Lattice ge-ometric characterization, analyzes binary representations, establishes the 50% duty cycle theorem, and explores connections to computer science and combinatorics. Suitable for motivated high school students through undergraduate mathematics majors. Total length: approximately 40 pages with appendice