

A307611


An Ackermannlike function arising from a puzzle by Hans Zantema.


1




OFFSET

1,2


COMMENTS

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 <= n1) and place two coins in B(i+1); (2) remove a coin from a nonempty B(i) (i <= n2) and switch the contents of B(i+1) and B(i+2).
The derivation and proof of the general formula involving a sequence of uparrows is by Richard Stong, Dan Velleman, and Stan Wagon.
The next term is too large to include (2^65537, it has 19729 digits).


REFERENCES

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


LINKS

Table of n, a(n) for n=1..5.
Wikipedia, Knuth's uparrow notation


FORMULA

Let f_n(x) = 2↑↑...↑x, with n Knuth uparrows, 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_(n2)(1).


EXAMPLE

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.


MATHEMATICA

f[n_][x_] := If[n == 0, 2x, Nest[f[n1], 1, x]]
F[n_] := Composition @@ (f /@ Range[0, n])
a[n_] := If[n <= 1, n, F[n2][1]]


CROSSREFS

Cf. A281701.
Sequence in context: A018355 A100083 A151406 * A053147 A128055 A061285
Adjacent sequences: A307608 A307609 A307610 * A307612 A307613 A307614


KEYWORD

nonn


AUTHOR

Stan Wagon, Apr 18 2019


STATUS

approved



