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%I #32 Mar 17 2021 19:49:54
%S 0,2,4,6,8,10,12,14,15,17,19,21,23,25,27,28,30,32,34,36,38,40,41,43,
%T 45,47,49,51,53,54,56,58,60,62,64,66,68,69,71,73,75,77,79,81,82,84,86,
%U 88,90,92,94,95,97,99,101,103,105,107,108,110,112,114,116,118,120,121
%N The 'Honeycomb' or 'Beehive' sequence: a(n) = ceiling(12^(1/4)*n).
%C Sequence gives the approximate perimeter of n hexagons of unit area, assuming each side is shared with another hexagon.
%C This could form a Beatty sequence.
%C This is not A195125, although they match until 37th term and never after the 1000th term.
%H ABC Shop, <a href="http://shop.abc.net.au/products/code-the-dvd">The Code with description</a>
%H American Scientist, <a href="http://www.americanscientist.org/issues/feature/the-soap-film-an-analogue-computer-/1">The Soap Film: An Analogue Computer</a>
%H American Scientist, <a href="http://www.americanscientist.org/issues/feature/2000/2/foams-and-honeycombs/1">Foams and Honeycombs</a>
%H The BBC Productions, <a href="http://www.bbc.co.uk/programmes/b00zs6sl">The Code</a>; see in particular <a href="https://www.youtube.com/watch?v=F5rWmGe0HBI">Behind the Beehive - The Code - Episode 2</a>
%H Thomas C. Hales, <a href="https://arxiv.org/abs/math/9906042">The Honeycomb Conjecture</a>, arXiv:math/9906042 [math.MG], 1999-2002.
%H Marcus P. F. du Sautoy, <a href="http://www.athenalearning.com">The Code</a>, a DVD produced by Athena, 2011.
%H Scienceray, <a href="http://scienceray.com/mathematics/is-bee-a-mathematician-mathematics-in-nature/">Is Bee a Mathematician? - mathematics in nature.</a>
%F a(n) = ceiling( 12^(1/4) * n ).
%t f[n_] := Ceiling[ Sqrt[ Sqrt[12]]*n]; Array[f, 66, 0]
%o (Python)
%o from sympy import integer_nthroot
%o def A228295(n): return 0 if n == 0 else 1+integer_nthroot(12*n**4,4)[0] # _Chai Wah Wu_, Mar 17 2021
%Y Cf. A011009.
%K nonn
%O 0,2
%A Marcus du Sautoy and _Robert G. Wilson v_, Aug 19 2013
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