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A348149
Variation of the Barnyard sequence A347581: a(n) is the minimum number of unit-length line segments required to enclose areas of 1 through n on a square grid when the number of segments is minimized as each area of incrementing size, starting at 1, is added.
1
4, 9, 14, 20, 26, 33, 40, 48, 55, 64
OFFSET
1,1
COMMENTS
In this variation of A347581 the areas must be added in the order of their sizes, from 1 through n, and as each area is added the minimum possible number of line segments must be used. This forces, for example, the first three areas of size 1, 2 and 3 to form a 2 X 3 block and thus they can never appear in any other arrangement in the final area. This is also true for n up to at least 9 due to the restriction of maximizing the usable edges for the next area. This leads to a(8) and a(10) containing one more line segment than the optimal solutions of A347581.
LINKS
Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2015.
EXAMPLE
Examples of n = 1 to n = 10 are given below. Note that for a(3) the configuration could also consist of the area of size 1 sitting above the area of size 2 with the area of size 3 forming an L-shaped block creating the minimal 2 X 3 block.
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|__| a(1) = 4
__ __ __
|__|__ __| a(2) = 9
__ __ __
|__|__ __| a(3) = 14
|__ __ __|
__ __ __
|__|__ __|
|__ __ __| a(4) = 20
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|__ __|
__ __ __
|__|__ __|__
|__ __ __| | a(5) = 26
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|__ __|__ __|
__ __ __
|__|__ __|__ __ __
|__ __ __| | | a(6) = 33
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|__ __|__ __|__ __|
__ __ __ __
__ __|__ |
|__|__ __|__ __ __|
|__ __ __| | | a(7) = 40
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|__ __|__ __|__ __|
__ __ __ __
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|__ __ __ __|
__ __|__ |
|__|__ __|__ __ __| a(8) = 48
|__ __ __| | |
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|__ __|__ __|__ __|
__ __ __ __ __ __ __
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| |__ __ __ __|
|__ __ __|__ |
|__|__ __|__ __ __| a(9) = 55
|__ __ __| | |
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|__ __|__ __|__ __|
__ __ __ __ __ __ __
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| |__ __ __ __|
__|__ __ __|__ |
| |__|__ __|__ __ __| a(10) = 64
| |__ __ __| | |
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| |__ __|__ __|__ __|
|__ __|
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KEYWORD
nonn,more
AUTHOR
Scott R. Shannon, Oct 03 2021
STATUS
approved