A052200 - OEIS

A052200

Number of n-state, 2-symbol, d+ in {LEFT, RIGHT}, 5-tuple (q, s, q+, s+, d+) (halting or not) Turing machines.

1, 64, 20736, 16777216, 25600000000, 63403380965376, 232218265089212416, 1180591620717411303424, 7958661109946400884391936, 68719476736000000000000000000, 739696442014594807059393047166976, 9711967541295580042210555933809967104, 152784834199652075368661148843397208866816

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OFFSET

0,2

COMMENTS

T in T_(n, k) is a Turing machine with n states and k symbols;

States q, q+ in set Q_n of n distinct states (plus the Halt state;)

Symbols s, s+ in set S_k of k distinct symbols (0 as the blank symbol;)

Shift direction d+ in {LEFT, RIGHT} (NONE is excluded here;)

+ suffix meaning next and q+ = f(q, s), s+ = g(q, s), d+ = h(q, s).

This sequence is computable. On the other hand, the busy beaver numbers are noncomputable, found only by examining each of the many n-state Turing machine programs' halting. - Michael Joseph Halm, Apr 29 2003

From Daniel Forgues, Jun 06 2011: (Start)

RE: Busy beaver halting Turing machines:

H in H_(n, k) is a halting* Turing machine with n states and k symbols;

* (on a blank tape, all 0's, as input)

sigma(H) is the number of non-blank symbols left on the tape by H;

s(H) is the number of steps (or shifts in our case) taken by H;

Sigma(n, k) = max {sigma(H) : H is a halting Turing machine with n states and k symbols}

S(n, k) = max {s(H) : H is a halting Turing machine with n states and k symbols}

Cf. A028444 for Sigma(n, 2), A060843 for S(n, 2). (End)

This sequence also counts machines with unreachable states, and all (up to (n-1)!) permutations of non-start states, which don't affect behavior. In contrast, A107668 only counts state graphs reaching all n states in canonical order. - John Tromp, Oct 17 2024

LINKS

Table of n, a(n) for n=0..12.

J. P. Jones, Recursive undecidability - an exposition, Amer. Math. Monthly, 81 (1974), 724-738.

Michael Somos, The Busy Beaver Problem

OEIS Wiki, Busy Beaver numbers

Index entries for sequences related to Busy Beaver problem

FORMULA

a(n) = (4*(n+1))^(2*n).

EXAMPLE

a(1) = 64 = (4*1+4)^(2*1) = 8^2.

MATHEMATICA

A052200[n_]:=(4(n+1))^(2n); Array[A052200, 20, 0] (* Paolo Xausa, May 21 2023 *)