Original entry on oeis.org
1, 1, 1, 6, 11, 35, 73, 182, 374, 816, 1615, 3244, 6160, 11678, 21353, 38742, 68541, 120082, 206448, 351386, 589237, 978626, 1605582, 2610694, 4201319, 6705559, 10607058, 16652362, 25937765, 40122446, 61629301, 94066442, 142668403
Offset: 0
Original entry on oeis.org
1, 2, 4, 11, 28, 65, 147, 321, 678, 1382, 2738, 5289, 9985, 18452, 33455, 59616, 104556, 180690, 308058, 518648, 863037, 1420480, 2314170, 3734063, 5970888, 9466452, 14887746, 23235296, 36000876, 55395893, 84680624, 128636339, 194239572
Offset: 0
A027709
Minimal perimeter of polyomino with n square cells.
Original entry on oeis.org
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
Jonathan Custance (jevc(AT)atml.co.uk)
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.
- 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.
- 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
Number of such polyominoes is in
A100092.
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a027709 0 = 0
a027709 n = a027434 n * 2 -- Reinhard Zumkeller, Mar 23 2013
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[2*Ceiling(2*Sqrt(n)): n in [0..100]]; // Vincenzo Librandi, May 11 2015
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interface(quiet=true); for n from 0 to 100 do printf("%d,", 2*ceil(2*sqrt(n))) od;
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Table[2*Ceiling[2*Sqrt[n]], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)
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from math import isqrt
def A027709(n): return 1+isqrt((n<<2)-1)<<1 if n else 0 # Chai Wah Wu, Jul 28 2022
Edited by Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002
A342243
Triangle T(n,p) read by rows: the number of n-celled polyominoes with perimeter 2p, 2 <= p <= 1+n.
Original entry on oeis.org
1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 0, 1, 11, 0, 0, 0, 1, 7, 27, 0, 0, 0, 0, 4, 21, 83, 0, 0, 0, 0, 2, 21, 91, 255, 0, 0, 0, 0, 1, 9, 89, 339, 847, 0, 0, 0, 0, 0, 6, 67, 393, 1360, 2829, 0, 0, 0, 0, 0, 1, 45, 325, 1713, 5255, 9734, 0, 0, 0, 0, 0, 1, 23, 275
Offset: 1
The triangle has rows n=1,2,3,... and columns p=2,3,4,5,...:
1;
0, 1;
0, 0, 2;
0, 0, 1, 4;
0, 0, 0, 1, 11;
0, 0, 0, 1, 7, 27;
0, 0, 0, 0, 4, 21, 83;
0, 0, 0, 0, 2, 21, 91, 255;
0, 0, 0, 0, 1, 9, 89, 339, 847;
0, 0, 0, 0, 0, 6, 67, 393, 1360, 2829;
0, 0, 0, 0, 0, 1, 45, 325, 1713, 5255, 9734;
...
A275937
The number of distinct patterns of the smallest number of unit squares required to enclose n units of area, where corner contact is allowed.
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 1, 4, 2, 4, 1
Offset: 0
Showing 1-5 of 5 results.
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