A027709 Minimal perimeter of polyomino with n square cells.

0, 4, 6, 8, 8, 10, 10, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 34

Offset: 0

Author

Jonathan Custance (jevc(AT)atml.co.uk)

Keywords

• easy
• nonn

Examples

			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.
		

References

• F. Harary and H. Harborth, Extremal Animals, Journal of Combinatorics, Information & System Sciences, Vol. 1, No 1, 1-8 (1976).
• W. C. Yang, Optimal polyform domain decomposition (PhD Dissertation), Computer Sciences Department, University of Wisconsin-Madison, 2003.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
• Henri Picciotto, Geometry Labs, Labs 8.1-8.3.
• J. Yackel, R. R. Meyer and I. Christou, Minimum-perimeter domain assignment, Mathematical Programming, vol. 78 (1997), pp. 283-303.
• Jason R. Zimba, Solution to Perimeter Problem, Jan 23 2015

Crossrefs

Cf. A000105, A067628 (analog for triangles), A075777 (analog for cubes).

Cf. A135711.

Number of such polyominoes is in A100092.

Programs

Haskell

a027709 0 = 0
a027709 n = a027434 n * 2  -- Reinhard Zumkeller, Mar 23 2013

Magma

[2*Ceiling(2*Sqrt(n)): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Maple

interface(quiet=true); for n from 0 to 100 do printf("%d,", 2*ceil(2*sqrt(n))) od;

Mathematica

Table[2*Ceiling[2*Sqrt[n]], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)

Python

from math import isqrt
def A027709(n): return 1+isqrt((n<<2)-1)<<1 if n else 0 # Chai Wah Wu, Jul 28 2022

Formula

a(n) = 2*ceiling(2*sqrt(n)).

a(n) = 2*A027434(n) for n > 0. - Tanya Khovanova, Mar 04 2008

Extensions

Edited by Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002