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A307611 An Ackermann-like function arising from a puzzle by Hans Zantema. 1
1, 2, 4, 8, 32 (list; graph; refs; listen; history; text; internal format)



a(n) is the largest number of coins obtainable by making repeated moves in this puzzle: Start with empty boxes B(i), i=1..n, and place one coin in B(1). One can iterate moves of two types: (1) remove a coin from a nonempty B(i) (i <= n-1) and place two coins in B(i+1); (2) remove a coin from a nonempty B(i) (i <= n-2) and switch the contents of B(i+1) and B(i+2).

The derivation and proof of the general formula involving a sequence of up-arrows is by Richard Stong, Dan Velleman, and Stan Wagon.

The next term is too large to include (2^65537, it has 19729 digits).


Dan Velleman and Stan Wagon, Bicycle or Unicycle?, MAA Press, to appear.


Table of n, a(n) for n=1..5.

Wikipedia, Knuth's up-arrow notation


Let f_n(x) = 2↑↑...↑x, with n Knuth up-arrows, so f_0(x) = 2x,

  f_1(x) = 2^x, f_2(x) = 2↑↑x = 2^2^...^2 with x copies of 2, etc. Let

  F_n be the composition of f_0, f_1,...,f_n. Then a(n) = F_(n-2)(1).


a(6) = f_0(f_1(f_2(f_3(f_4(1))))) = f_0(f_1(f_2(f_3(2))))

      = f_0(f_1(f_2(4))) = f_0(f_1(65536)) = f_0(2^65536) = 2^65537.


f[n_][x_] := If[n == 0, 2x, Nest[f[n-1], 1, x]]

F[n_] := Composition @@ (f /@ Range[0, n])

a[n_] := If[n <= 1, n, F[n-2][1]]


Cf. A281701.

Sequence in context: A018355 A100083 A151406 * A053147 A128055 A061285

Adjacent sequences:  A307608 A307609 A307610 * A307612 A307613 A307614




Stan Wagon, Apr 18 2019



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Last modified January 16 00:05 EST 2021. Contains 340195 sequences. (Running on oeis4.)