login
A089071
Number of liberties a big eye of size n gives in the game of Go.
14
1, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380
OFFSET
1,2
COMMENTS
The terms after the seventh are considered to be of only theoretical importance, since the largest dead shape is six spaces.
FORMULA
a(n) = a(n-1) + n - 2 for n>=3.
From Paul Barry, Dec 07 2009: (Start)
G.f.: (1 - x + x^3)/(1-x)^3.
a(n) = n + 1 - 0^n + C(n-1,2). (End)
a(n) = A022856(n+2). - R. J. Mathar, Oct 30 2011
EXAMPLE
A 5-space big eye can be almost filled in 4 moves, after which one takes and has a 4-space big eye (5 liberties) left. This gives a total of 4 + 5 moves for the opponent and 1 for oneself, for de facto 8 liberties.
MATHEMATICA
Join[{1}, Binomial[Range[65], 2] +2] (* G. C. Greubel, Oct 31 2022 *)
PROG
(Magma) [n eq 1 select 1 else Binomial(n-1, 2) +2: n in [1..65]]; // G. C. Greubel, Oct 31 2022
(SageMath) [binomial(n-1, 2)+2-int(n==1) for n in range(1, 65)] # G. C. Greubel, Oct 31 2022
CROSSREFS
Cf. A022856.
Sequence in context: A338921 A333343 A022856 * A177205 A275580 A175829
KEYWORD
nonn,easy
AUTHOR
André Engels, Dec 03 2003
EXTENSIONS
More terms from David Wasserman, Aug 29 2005
STATUS
approved