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 A208229 #34 Jul 13 2025 16:44:51
%S A208229 3,3,15,19,37,51,69,97,111,161,167,241,247,327,341,435,451,547,571,
%T A208229 665,709,801,849
%N A208229 Consider Wolfram's universal 2-state 3-symbol Turing machine on a one-way-infinite tape with all but the first n cells initially blank and the head initially in state 1. a(n) is the maximum number of steps the Turing machine can make before halting.
%C A208229 The head starts in state 1 on the leftmost cell (cell 1.) The machine halts if the head moves to the left of the first cell. a(n) gives the maximum halting time for any of the 3^n initial configurations of the first n cells.
%C A208229 Because the rule (rule 596440) is universal, the halting problem is undecidable for arbitrary tapes. However, it is not known whether universal computation can be achieved with a finite number of nonzero starting cells. Thus, this function might be computable.
%C A208229 Since the head starts in state one, it will immediately move left and halt unless the first cell starts out with a 0.
%C A208229 For initial conditions with n<=11, the machine always halts (the Mathematica code given assumes that the machine halts for all finite configurations). Whether this remains true is an open question.
%D A208229 Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 709, 1120.
%H A208229 Michael S. Branicky, <a href="/A208229/a208229.py.txt">Python program for OEIS A208229</a>
%H A208229 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wolfram_2,3_Turing_Machine">Wolfram's 2-state 3-symbol Turing machine</a>
%H A208229 Stephen Wolfram, <a href="http://www.wolframscience.com/prizes/tm23/">Wolfram 2,3 Turing Machine Research Prize</a>
%e A208229 For n=3, the initial tape 021000000... runs for 19 steps, before terminating in the state 22221200000... with the head one cell to the left of the tape. This is the longest running time without starting with nonzero symbols further to the right, so a(3)=19
%t A208229 r = {{1, 2} -> {1, 1, -1}, {1, 1} -> {1, 2, -1}, {1, 0} -> {2, 1, 1}, {2, 2} -> {1, 0, 1}, {2, 1} -> {2, 2, 1}, {2, 0} -> {1, 2, -1}}; len[n_, k_] := Length[NestWhileList[TuringMachine[r, #] &, {{1,2}, {Prepend[IntegerDigits[k, 3, n], 3], 0}}, UnsameQ, All]] - 2; a[n_] := Max[Table[len[n, k], {k, 0, 3^(n - 1) - 1}]]; Join[{3},Table[a[n],{n,1,8}]]
%o A208229 (Python) # see Branicky link
%K A208229 nonn,more
%O A208229 0,1
%A A208229 _Ben Branman_, Jan 10 2013
%E A208229 a(12)-a(14) from _Robert G. Wilson v_, Mar 22 2015
%E A208229 a(15)-a(22) from _Michael S. Branicky_, Jul 07 2025