This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337025 #37 Jun 29 2025 22:43:57 %S A337025 1,16,4096,2985984,4294967296,10240000000000,36520347436056576, %T A337025 182059119829942534144,1208925819614629174706176, %U A337025 10314424798490535546171949056,109951162777600000000000000000000,1432052311740255546466984939315265536 %N A337025 Number of n-state 2-symbol halt-free Turing machines. %C A337025 A Turing machine is halt-free if none of its instructions lead to the halt state. %C A337025 This sequence is strictly less than A052200(n) for all n > 0, since halt-free n-state machines are a strict subset of all n-state machines. %C A337025 Solutions to the so-called "Beeping Busy Beaver" problem will almost certainly be halt-free programs. %C A337025 For n > 1, a(n) is the absolute value of the determinant of a Hadamard matrix of size 4(n-1). This is apparent from the fact that HH^T = nI_n implies det(H)^2 = det(nI_n) = n^n, so det(H) = +- n^(n/2) and plugging in 4n yields a(n). - _Nour Abouyoussef_, Jun 25 2025 %H A337025 Scott Aaronson, <a href="https://www.scottaaronson.com/papers/bb.pdf">The Busy Beaver Frontier</a>. %H A337025 Nick Drozd, <a href="https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html">Beeping Busy Beavers</a>. %F A337025 a(n) = ((4*n)^2)^n. %o A337025 (Python) [((4 * n) ** 2) ** n for n in range(12)] %Y A337025 Cf. A052200. %K A337025 nonn,easy %O A337025 0,2 %A A337025 _Nicholas Drozd_, Aug 11 2020