OEIS A000336: Product recurrence and Hasler's elegant identity

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.

Note:  This podcast was AI-generated, and sometimes AI can make mistakes.  Please double-check any critical information.

Sponsored by Embersilk LLC