A332755 Lapidary numbers.

1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 31, 45, 61, 87, 119, 171, 233, 334, 459, 655, 904, 1288, 1782, 2535, 3517, 4995, 6935, 9848, 13703, 19437, 27070, 38376, 53528, 75842, 105878, 149966, 209555, 296707, 414922, 587304, 821853, 1163052, 1628574, 2304082

Offset

0,5

Comments

Consider a two-player stone-throwing game with a single shared pile of stones. The players alternately remove one or more stones from the pile until it is empty. In addition, each player seeks to communicate a message through their sequence of moves. If there are initially n stones then a(n) is the largest number m such that both players can communicate at least m distinct messages.

For n > 0, a(n) is also the size of the Durfee square of the partition defined in A064660.

Links

Peter J. Taylor, Table of n, a(n) for n = 0..60

Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones (preprint).

Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones, Eureka, 65 (2018), pp. 89-90.

Peter J. Taylor, Python program

Formula

Asymptotically, a(n) is within a subexponential factor of 2^(n/2).

Example

For n=4, one strategy which allows both players to communicate one of two messages is each remove one or two stones on their first turn.

Crossrefs

Cf. A064660.

Sequence in context: A362048 A334626 A339235 * A017912 A102543 A173383

Adjacent sequences: A332752 A332753 A332754 * A332756 A332757 A332758

Keyword

nonn

Author

Peter J. Taylor, Feb 22 2020

Status

approved