A089071 Number of liberties a big eye of size n gives in the game of Go.

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

Author

André Engels, Dec 03 2003

Keywords

• nonn
• easy

Comments

The terms after the seventh are considered to be of only theoretical importance, since the largest dead shape is six spaces.

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Senseis Library, Big Eye Liberties
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A022856.

Programs

Magma

[n eq 1 select 1 else Binomial(n-1,2) +2: n in [1..65]]; // G. C. Greubel, Oct 31 2022

Mathematica

Join[{1}, Binomial[Range[65],2] +2] (* 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

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

Extensions

More terms from David Wasserman, Aug 29 2005