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

0,2

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

Formula

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

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A135711.

Number of such polyominoes is in A100092.

Sequence in context: A163639 A196355 A095253 * A196358 A079775 A247654

Adjacent sequences: A027706 A027707 A027708 * A027710 A027711 A027712

Keyword

easy,nonn

Author

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

Extensions

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

Status

approved