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

%I A089071 #21 Nov 01 2022 03:08:45
%S A089071 1,2,3,5,8,12,17,23,30,38,47,57,68,80,93,107,122,138,155,173,192,212,
%T A089071 233,255,278,302,327,353,380,408,437,467,498,530,563,597,632,668,705,
%U A089071 743,782,822,863,905,948,992,1037,1083,1130,1178,1227,1277,1328,1380
%N A089071 Number of liberties a big eye of size n gives in the game of Go.
%C A089071 The terms after the seventh are considered to be of only theoretical importance, since the largest dead shape is six spaces.
%H A089071 G. C. Greubel, <a href="/A089071/b089071.txt">Table of n, a(n) for n = 1..1000</a>
%H A089071 Senseis Library, <a href="http://senseis.xmp.net/?BigEyeLiberties">Big Eye Liberties</a>
%H A089071 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A089071 a(n) = a(n-1) + n - 2 for n>=3.
%F A089071 From _Paul Barry_, Dec 07 2009: (Start)
%F A089071 G.f.: (1 - x + x^3)/(1-x)^3.
%F A089071 a(n) = n + 1 - 0^n + C(n-1,2). (End)
%F A089071 a(n) = A022856(n+2). - _R. J. Mathar_, Oct 30 2011
%e A089071 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.
%t A089071 Join[{1}, Binomial[Range[65],2] +2] (* _G. C. Greubel_, Oct 31 2022 *)
%o A089071 (Magma) [n eq 1 select 1 else Binomial(n-1,2) +2: n in [1..65]]; // _G. C. Greubel_, Oct 31 2022
%o A089071 (SageMath) [binomial(n-1,2)+2-int(n==1) for n in range(1,65)] # _G. C. Greubel_, Oct 31 2022
%Y A089071 Cf. A022856.
%K A089071 nonn,easy
%O A089071 1,2
%A A089071 _André Engels_, Dec 03 2003
%E A089071 More terms from _David Wasserman_, Aug 29 2005