A261491 a(n) = ceiling(2 + sqrt(8*n-4)).

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, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 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

Offset: 1

Author

Juhani Heino, Aug 21 2015

Keywords

• nonn
• easy

Comments

Conjecture: a(n) = minimal number of stones needed to surround area n in the middle of a Go board (infinite if needed).

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.

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.

The opposite of this would be an area of 1 X n; it demands the maximal number of stones, a(n) = 2 + 2n.

Equivalently, a(n) is the minimum (cell) perimeter of any polyomino of n cells. - Sean A. Irvine, Oct 17 2020

Examples

			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.
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.
___D___
__A5C__
_B104E_
G30007J
_F206I_
__H8K__
___L___
		

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A001971.

Programs

Mathematica

Array[Ceiling[2 + Sqrt[8 # - 4]] &, {86}] (* Michael De Vlieger, Oct 23 2015 *)

PARI

a(n)=sqrtint(8*n-5)+3 \\ Charles R Greathouse IV, Aug 21 2015

Formula

a(n) = ceiling(2 + sqrt(8*n-4)).

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