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%I #25 Jun 06 2023 03:09:00
%S 4,9,14,20,26,33,40,48,55,64
%N 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.
%C 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.
%H Sascha Kurz, <a href="https://arxiv.org/abs/math/0506428">Counting polyominoes with minimum perimeter</a>, arXiv:math/0506428 [math.CO], 2015.
%e 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.
%e .
%e __
%e |__| a(1) = 4
%e __ __ __
%e |__|__ __| a(2) = 9
%e __ __ __
%e |__|__ __| a(3) = 14
%e |__ __ __|
%e __ __ __
%e |__|__ __|
%e |__ __ __| a(4) = 20
%e | |
%e |__ __|
%e __ __ __
%e |__|__ __|__
%e |__ __ __| | a(5) = 26
%e | | |
%e |__ __|__ __|
%e __ __ __
%e |__|__ __|__ __ __
%e |__ __ __| | | a(6) = 33
%e | | | |
%e |__ __|__ __|__ __|
%e __ __ __ __
%e __ __|__ |
%e |__|__ __|__ __ __|
%e |__ __ __| | | a(7) = 40
%e | | | |
%e |__ __|__ __|__ __|
%e __ __ __ __
%e | |
%e |__ __ __ __|
%e __ __|__ |
%e |__|__ __|__ __ __| a(8) = 48
%e |__ __ __| | |
%e | | | |
%e |__ __|__ __|__ __|
%e __ __ __ __ __ __ __
%e | | |
%e | |__ __ __ __|
%e |__ __ __|__ |
%e |__|__ __|__ __ __| a(9) = 55
%e |__ __ __| | |
%e | | | |
%e |__ __|__ __|__ __|
%e __ __ __ __ __ __ __
%e | | |
%e | |__ __ __ __|
%e __|__ __ __|__ |
%e | |__|__ __|__ __ __| a(10) = 64
%e | |__ __ __| | |
%e | | | | |
%e | |__ __|__ __|__ __|
%e |__ __|
%e .
%Y Cf. A347581, A001168, A291808, A291809, A328020, A291806, A006983.
%K nonn,more
%O 1,1
%A _Scott R. Shannon_, Oct 03 2021
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