

A094777


Number of legal positions in Go played on an n X n grid (each group must have at least one liberty).


11



1, 57, 12675, 24318165, 414295148741, 62567386502084877, 83677847847984287628595, 990966953618170260281935463385, 103919148791293834318983090438798793469, 96498428501909654589630887978835098088148177857, 793474866816582266820936671790189132321673383112185151899, 57774258489513238998237970307483999327287210756991189655942651331169, 37249792307686396442294904767024517674249157948208717533254799550970595875237705, 212667732900366224249789357650440598098805861083269127196623872213228196352455447575029701325
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OFFSET

1,2


COMMENTS

John Tromp wrote a small C program to compute the number for boards up to size 4 X 5, given in the rec.games.go posting below. Gunnar Farnebaeck (gunnar(AT)lysator.liu.se) wrote a pike script to compute the number by dynamic programming, which handles sizes up to 12 X 12 (available upon request).


LINKS

British Go Association, Go


FORMULA

3^(n*n) is a trivial upper bound.
Tromp & Farnebäck prove that a(n) = (1 + o(1)) * L^(n^2), and conjecture that a(n) ~ A * B^(2n) * L^(n^2) * (1 + O(n*p^n)) for some constants A, B, L, and p < 1.  Charles R Greathouse IV, Feb 08 2016


EXAMPLE

The illegal 2 X 2 positions are the 2^4 with no empty points and the 4*2 having a stone adjacent to 2 opponent stones that share a liberty. That leaves 3^4168 = 57 legal positions.


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

Michal Koucky should be credited for carrying most of the computational load for computing the n=14, 15 and 16 results with his filebased implementation.


STATUS

approved



