In this Deep Dive we unpack OEIS A000338. We explore its generating function, explain what the offset (offset 3, 1) means, and show how the infinite power-series expansion yields the sequence beginning 5, 18, 42, 75, 117. We derive the linear recurrence and connect the terms to a combinatorial story about discordant permutations, as discussed in J. Reordan's 1954 work and traced back to N. J. Sloan’s 1973 Handbook of Integer Sequences. The episode illustrates how a compact algebraic form hides a rich mathematical landscape of counting with forbidden positions.

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A guided tour through what mathematicians call beautiful—from Euler’s identity and Fermat’s theorem to Cantor’s diagonal argument and visual proofs. We’ll explore how beauty arises in elegant results, clever proofs, or even abstract structures, and what neuroscience reveals about this universal sense of harmony.

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We follow Cayley’s transform from real homographies to complex disk models, mapping skew-symmetric matrices to unitary rotations, extending to quaternions, and finally to operators on Hilbert spaces. A single idea that tames infinity, links Poincaré models, and even finds engineering use in the Smith chart.

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Anton Korinek's NBER working paper argues that AI is evolving from responsive tools to autonomous agents that can plan, run multi-step analyses, and collaborate across tools to advance economic research. We trace the arc from System 1 to System 2 reasoning to agentic AI, explore vibe coding, private-data protocols, and cost/governance issues, and discuss why human judgment remains essential even as AI reshapes the questions we can tackle.

Source: https://www.nber.org/papers/w34202

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In this Deep Dive, we explore A000337, a small-seeming sequence that threads through binary arithmetic, geometry, and number theory. We'll trace how its simple definition links to counting zeros and bits in binary lists, to directed column-convex polyominoes, and to the genus of cube graphs. Along the way we’ll encounter prime and semiprime patterns noted by researchers, a neat generating function and a linear recurrence, and a combinatorial interpretation as sums of subset maxima. Join us as we uncover how one formula can map across disparate areas of math—and consider what other seemingly plain sequences in the OEIS are quietly weaving together multiple fields.

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From its 2005 discovery by Mike Brown's team to its high-inclination orbit that kept it hidden in dense star fields, Makemake is a bright but enigmatic dwarf planet beyond Neptune. We explore how its Easter Island name origin became Makemake, its red, methane- and ethane-ice surface at about 40 K, and the surprising hints that it may host geothermal activity and a subsurface ocean. We also examine the 2011 occultation showing no substantial atmosphere, the single moon MK2, and what Makemake teaches us about the diversity of the Kuiper Belt and how internal heat reshapes our ideas of habitability in the outer solar system.

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In this Deep Dive we zoom in on OEIS A000336, the classic product-recurrence sequence. Starting with a1=1, a2=2, a3=3, a4=4 and, for n≥5, an = an−1 · an−2 · an−3 · an−4, the seeds explode into astonishing growth: a5=24, a6=576, a7=165,888, a8=9,172,942,848, and far beyond. We’ll unpack why such a simple rule yields such rapid, almost astronomical expansion and how later terms acquire hundreds of digits (a12, for example, reaches 139 digits). A standout twist is Hasler’s elegant alternative formula: for n≥6, an = (an−1)² / an−5. This identity reveals a telescoping structure hiding in the product recurrence and helps explain the hidden regularity behind the chaos. We’ll also touch on how A000336 connects to other OEIS sequences—e.g., exponent-tracking sequences like A001631 for the 3-adic/prime-exponent footprint of an—and show how cross-references and code examples (Maple, Mathematica, PRI) illuminate these relationships. The episode concludes with reflections on how such deceptively simple rules can yield deep, structured mathematics and what they hint at for exploring other OEIS entries.

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A guided tour of how the book evolved—from Mesopotamian clay tablets and Egyptian scrolls to the codex, movable type, steam presses, libraries, ISBNs, and the Kindle era. We trace the social and technological shifts that made books portable, accessible, and influential—and how censorship, accessibility, and digital media continue to reshape the way we record and share knowledge.

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We explore Albania's audacious move to appoint Diella, an AI minister tasked with policing public procurement and promising 100% corruption-free tenders. The episode digs into the tech, governance, legal, and geopolitical implications—examining accountability, transparency, and what this bold experiment means for Albania's EU ambitions and the future of political trust in a digitizing world.

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Imagine you must rotate a line segment through every direction in the smallest possible space. The Kakeya problem began in 1917, provoking Besicovitch’s startling zero-area sets and a shift from area to dimension via Minkowski dimension. We trace the arc from intuitive puzzles to counterintuitive constructions—Perron trees, Paul joins, and the polynomial method—including the finite-field version and its famous resolution. In March 2025, Hong Wang and Joshua Azal announced a complete solution in three dimensions, a milestone with deep implications for higher dimensions and analysis. We unpack the ideas, the breakthroughs, and what lies ahead for n ≥ 4—and connections to physics and computation.

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