

A261491


a(n) = ceiling(2 + sqrt(8*n4)).


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
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OFFSET

1,1


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+(x1)^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(8n4))/4. And since a(n)=4x, the formula a(n) = 2 + sqrt(8n4) 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.


LINKS

Table of n, a(n) for n=1..86.
Kival Ngaokrajang, Illustration of initial terms


FORMULA

a(n) = ceiling(2 + sqrt(8*n4)).


EXAMPLE

Start with the 5cell 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___


MATHEMATICA

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


PROG

(PARI) a(n)=sqrtint(8*n5)+3 \\ Charles R Greathouse IV, Aug 21 2015


CROSSREFS

Cf. A001971.
Sequence in context: A300707 A296183 A021876 * A005670 A234948 A123860
Adjacent sequences: A261488 A261489 A261490 * A261492 A261493 A261494


KEYWORD

nonn,easy


AUTHOR

Juhani Heino, Aug 21 2015


STATUS

approved



