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A027709 Minimal perimeter of polyomino with n square cells.

%I A027709 #56 Aug 06 2024 07:28:16
%S A027709 0,4,6,8,8,10,10,12,12,12,14,14,14,16,16,16,16,18,18,18,18,20,20,20,
%T A027709 20,20,22,22,22,22,22,24,24,24,24,24,24,26,26,26,26,26,26,28,28,28,28,
%U A027709 28,28,28,30,30,30,30,30,30,30,32,32,32,32,32,32,32,32,34,34,34,34,34,34
%N A027709 Minimal perimeter of polyomino with n square cells.
%D A027709 F. Harary and H. Harborth, Extremal Animals, Journal of Combinatorics, Information & System Sciences, Vol. 1, No 1, 1-8 (1976).
%D A027709 W. C. Yang, Optimal polyform domain decomposition (PhD Dissertation), Computer Sciences Department, University of Wisconsin-Madison, 2003.
%H A027709 Reinhard Zumkeller, <a href="/A027709/b027709.txt">Table of n, a(n) for n = 0..10000</a>
%H A027709 Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, <a href="https://doi.org/10.1007/s00026-022-00631-1">Extremal {p, q}-Animals</a>, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
%H A027709 Henri Picciotto, <a href="http://www.mathedpage.org/geometry-labs/">Geometry Labs</a>, Labs 8.1-8.3.
%H A027709 J. Yackel, R. R. Meyer and I. Christou, <a href="https://citeseerx.ist.psu.edu/pdf/4174091178f0abc27ab6404fa241575a1f16bdc4">Minimum-perimeter domain assignment</a>, Mathematical Programming, vol. 78 (1997), pp. 283-303.
%H A027709 Jason R. Zimba, <a href="http://jzimba.blogspot.com/2015/06/solution-to-perimeter-puzzle.html">Solution to Perimeter Problem</a>, Jan 23 2015
%F A027709 a(n) = 2*ceiling(2*sqrt(n)).
%F A027709 a(n) = 2*A027434(n) for n > 0. - _Tanya Khovanova_, Mar 04 2008
%e A027709 a(5) = 10 because we can arrange 5 squares into 2 rows, with 2 squares in the top row and 3 squares in the bottom row. This shape has perimeter 10, which is minimal for 5 squares.
%p A027709 interface(quiet=true); for n from 0 to 100 do printf("%d,", 2*ceil(2*sqrt(n))) od;
%t A027709 Table[2*Ceiling[2*Sqrt[n]], {n, 0, 100}] (* _Wesley Ivan Hurt_, Mar 01 2014 *)
%o A027709 (Haskell)
%o A027709 a027709 0 = 0
%o A027709 a027709 n = a027434 n * 2  -- _Reinhard Zumkeller_, Mar 23 2013
%o A027709 (Magma) [2*Ceiling(2*Sqrt(n)): n in [0..100]]; // _Vincenzo Librandi_, May 11 2015
%o A027709 (Python)
%o A027709 from math import isqrt
%o A027709 def A027709(n): return 1+isqrt((n<<2)-1)<<1 if n else 0 # _Chai Wah Wu_, Jul 28 2022
%Y A027709 Cf. A000105, A067628 (analog for triangles), A075777 (analog for cubes).
%Y A027709 Cf. A135711.
%Y A027709 Number of such polyominoes is in A100092.
%K A027709 easy,nonn
%O A027709 0,2
%A A027709 Jonathan Custance (jevc(AT)atml.co.uk)
%E A027709 Edited by Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002