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%I #29 Aug 19 2021 17:50:48
%S 4,6,7,8,8,9,10,10,11,11,12,12,12,13,13,14,14,14,15,15,15,16,16,16,16,
%T 17,17,17,18,18,18,18,19,19,19,19,20,20,20,20,20,21,21,21,21,22,22,22,
%U 22,22,23,23,23,23,23,24,24,24,24,24,24,25,25,25,25,25,26,26,26,26,26,26,27,27,27,27,27,27,28,28,28,28,28,28,28,29
%N a(n) = ceiling(2 + sqrt(8*n-4)).
%C Conjecture: a(n) = minimal number of stones needed to surround area n in the middle of a Go board (infinite if needed).
%C The formula was constructed this way: when the area is in a diamond shape with x^2+(x-1)^2 places, it can be surrounded by 4x stones. So, a(1)=4, a(5)=8, a(13)=12 etc.
%C The positive solution to the quadratic equation 2x^2 - 2x + 1 = n is x = (2 + sqrt(8n-4))/4. And since a(n)=4x, the formula a(n) = 2 + sqrt(8n-4) holds for the positions mentioned. But incredibly also the intermediate results seem to match when the ceiling function is used.
%C The opposite of this would be an area of 1 X n; it demands the maximal number of stones, a(n) = 2 + 2n.
%C Equivalently, a(n) is the minimum (cell) perimeter of any polyomino of n cells. - _Sean A. Irvine_, Oct 17 2020
%H Kival Ngaokrajang, <a href="/A261491/a261491.pdf">Illustration of initial terms</a>
%F a(n) = ceiling(2 + sqrt(8*n-4)).
%F For n > 2, a(n) - a(n-1) = 1 if n is of the form 2*(k^2+k+1), 2*k^2 + 1 or (k^2+k)/2 + 1, otherwise 0. - _Jianing Song_, Aug 10 2021
%e Start with the 5-cell area that is occupied by 0's and surrounded by stones 1..8. Add those surrounding stones to the area, one by one. At points 1, 2, 4 and 6, the number of surrounding stones is increased; elsewhere, it is not.
%e Next, do the same with stones A..L. At points A, C, F and I, the number of surrounding stones is increased; elsewhere, it is not.
%e ___D___
%e __A5C__
%e _B104E_
%e G30007J
%e _F206I_
%e __H8K__
%e ___L___
%t Array[Ceiling[2 + Sqrt[8 # - 4]] &, {86}] (* _Michael De Vlieger_, Oct 23 2015 *)
%o (PARI) a(n)=sqrtint(8*n-5)+3 \\ _Charles R Greathouse IV_, Aug 21 2015
%Y Cf. A001971.
%K nonn,easy
%O 1,1
%A _Juhani Heino_, Aug 21 2015
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