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A281701 a(n) is the largest number of coins obtainable by making repeated moves in this puzzle: Start with 1 coin in each of n boxes B(i), i=1..n. 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). 2
1, 3, 7, 28 (list; graph; refs; listen; history; text; internal format)



An Ackermann-like function. The underlying puzzle was invented by Hans Zantema. The derivation and proof of the general formula involving a palindromic sequence of up-arrows is by Richard Stong.

The next term is too large to include (2^16385, it has 4933 digits).


Table of n, a(n) for n=1..4.

Zuming Feng, Po-Shen Loh, and Yi Sun, 51st International Mathematical Olympiad, Math. Mag. 83 (2010), pp. 320-323.

Terence Tao, Minipolymath2 project: IMO 2010 Q5 (2010)

A. van den Brandhof, J. Guichelaar, and A. Jaspers, Half a Century of Pythagoras Magazine, MAA, 2015, 225

Stan Wagon, The Generous Automated Teller Machine

Stan Wagon, Richard Stong's proof of the uparrow formula

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-4).

Let G_n be the same composition but in the opposite order.

Then a(n) = G_n(F_n(7)), a formula due to Richard Stong.


a(5) = f_0(f_1(f_1(f_0(7)))) = 2*2^(2^(2*7)) = 2*2^(2^14) = 2^16385.


Cf. A307611.

Sequence in context: A280020 A232806 A270349 * A101303 A148751 A148752

Adjacent sequences:  A281698 A281699 A281700 * A281702 A281703 A281704




Stan Wagon, Jan 27 2017



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Last modified October 6 10:58 EDT 2022. Contains 357263 sequences. (Running on oeis4.)