Intellectually Curious

In this episode, we dive into Mikkel’s Pentagram Theorem: five circles arranged around a pentagram intersect such that the adjacent circles meet at a pentagram vertex, and the second intersection points all lie on one circle. We show how two starting viewpoints—from a pentagon extended into a star, or from circumcircles around the star’s tips—lead to the same elegant structure. We also discuss the converse: if the five circle centers are concyclic, the connecting lines of those second intersections form a new pentagram whose vertices sit on the original circles. Plus a touch of history and a visual intuition to glimpse the hidden order in geometry.

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We unpack an NBER paper arguing that stablecoins do more than money — they reshape global finance through a borderless digital economy called DECO. We explore reserve-backed vs crypto-backed designs, the two channels linking digital demand to dollars, and surprising implications for US rates, global borrowing, and consumption volatility—highlighting uneven impacts on Americans, the rest of the world, and the builders inside the DECO.

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We explore the intriguing class of integral‑sided triangles whose orthocenter and circumcenter stand in a precise one‑to‑two relationship at a vertex, using the 6‑7‑8 example as our anchor. We’ll define S_P, show how small cases like S_10 rise to S_100, and tackle the monumental S_107 (perimeters up to 10,000,000). The episode blends elegant geometry with heavy computation, discussing the algorithms and number‑theoretic ideas needed to sift billions of candidates and what such a search reveals about the choreography between shape and numbers, guided by a diagram and the paper Fortunate Triangles, Orthocenter and Circumcenter Distances.

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In this episode of The Deep Dive we explore A00291, the bipartite partitions sequence. We unpack its multiple lives: counting bipartitions of N white objects with two black ones; counting factorizations of P^N Q^2 with distinct primes; and relating to multiset partitions of a 2-element multiset. We also discuss connections to other fundamental OEIS sequences (like A000070 and A00097), the asymptotic growth formula, and the notable contributors—Sloan, Chima, Gupta, Knuth—whose work helped reveal the underlying structure. It’s a vivid example of how a simple counting problem can bridge combinatorics and number theory, inviting you to explore the rich web of links in the OEIS.

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Dive into the classic sequence of perfect squares, A000290. We’ll trace how n^2 shows up in elegant ways—like being the sum of consecutive odd numbers and the sum of two consecutive triangular numbers—and peek at the historical milestone of its first electronic-computer calculation (EDSCAC, Cambridge, May 6, 1949). We’ll connect squares to deeper math: why they have an odd number of divisors, their role in the Basel problem and the Riemann zeta function (zeta(2) = pi^2/6), and a curious universal pattern: for any even-length block of consecutive integers of length 2k, the difference between the sum of the second half and the first half is k^2. We’ll illustrate with a concrete example (61–70 yields 25) and reflect on how a seemingly simple sequence hides rich structure across number theory, history, and beyond.

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Welcome, curious minds. In this episode we dive into A000289, the nonlinear recurrence that rockets from simple beginnings to monstrous numbers, while keeping every pair of terms coprime. We unpack the defining rule a(n) = a(n-1)^2 - 3·a(n-1) + 3 (and the related alternate forms), trace its tame early terms—1, 4, 7, 31—and explore its astonishing growth and the “infinite coprime” property. We’ll place it in the broader family of mutual‑k residues with k = 3, linking it to famous sequences like Sylvester’s and a Fermat‑related cousin. Along the way we’ll peek at different representations—doubly exponential ceiling forms and a product-based recurrence a(n) = 3 + ∏_{i< n} a(i)—showing how the same object can be viewed through multiple mathematical lenses. Join us as we uncover the hidden structure, surprising connections, and the beauty of simple rules leading to deep number-theoretic order.

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In this Deep Dive, we unpack Rich Sutton's 2019 Bitter Lesson: over decades, AI breakthroughs favor general methods and scaled computation over hand-crafted knowledge. We trace examples from chess to Go to speech and vision, and discuss why meta-methods that learn and search at scale may be the key to future discovery.

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We explore A000288, the tetranacci-type sequence with starting values 1,1,1,1. Its simple recurrence a(n)=a(n−1)+a(n−2)+a(n−3)+a(n−4) leads to surprising structure: a combinatorial interpretation counting 4-letter words over {0,1,2,3} with the rule that each 1, 2, or 3 must be followed by at least one, two, or three zeros respectively; it also exhibits Benford-like leading-digit distribution and grows with the tetranacci constant around 1.92756. We’ll also touch on its generating function and what these properties reveal about the richness lurking in simple recurrences.

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We explore the messy, real-world 'garbage can' view of work and the bitter lesson from AI research: brute computation often outpaces carefully encoded knowledge. From process maps to autonomous AI agents (handcrafted vs outcome-trained), we ask whether we should map the mess or define outcomes and let AI learn the rest. And we consider the human side—trust, collaboration, and values AI struggles to quantify—as we probe how AI fits into the messy realities of modern work.

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We explore A000287, the number of rooted polyhedral graphs with n edges. Rooted means a distinguished edge on the polyhedral skeleton, so counting distinguishes shapes that would be equivalent without the root; the sequence begins with n = 6 → 1, n = 7 → 0, and then jumps to much larger values, with a striking parity pattern (odd exactly when n+2 is a power of two). We sketch the main tools used to study it—explicit recurrences (including Plouffe’s four-step recurrence), generating functions, and differential equations for the generating function—along with the large-n asymptotics that reveal connections to continuous analysis (involving constants like pi) behind this discrete counting problem.

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