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%I #48 Oct 19 2023 02:40:40
%S 1,1,1,2,2,3,3,4,5,6,6,7,8,8,9,10,10,11,12,13,13,14,15,15,16,17,17,18,
%T 19,20,20,21,22,22,23,24,25,25,26,27,27,28,29,30,30,31,32,32,33,34,34,
%U 35,36,37,37,38,39,39,40,41,42,42,43,44,44,45,46,46,47,48
%N Minimum integer length of a segment that touches the interior of n squares on a unit square grid.
%C This sequence, {a(n)}, is the "inverse" of A346232, {b(n)}, in the following sense: a(n) = min{L positive integer with b(L)>=n} and b(n) = max{S positive integer with a(S) <= n}.
%C The sequence is nondecreasing.
%C Except for the initial run of 3 equal values, it is formed by runs of 1 or 2 equal values, with an increment of 1 between consecutive runs.
%C There can be no more than 3 different consecutive terms.
%C A run of 2 equal values always has 2 different terms before and 2 different terms after the run, except for the initial terms (1, 1, 1, 2, 2, 3, 3).
%H Alex Arkhipov and Luis Mendo, <a href="https://doi.org/10.1112/mtk.12223">On the number of tiles visited by a line segment on a rectangular grid</a>, Mathematika, vol. 69, no. 4, pp. 1242-1281, October 2023. Also on <a href="https://arxiv.org/abs/2201.03975">arXiv</a>, arXiv:2201.03975 [math.MG], 2022-2023.
%F a(n) = 1 for n <= 3; a(n) = ceiling(sqrt((n-3)^2/2+1)) for n >= 4.
%e A segment of length 1 can touch a maximum of 3 squares (segment close to a square vertex and oriented at 45 degrees; see image in A346232), therefore a(1) = a(2) = a(3) = 1.
%e A segment of length 2 can touch a maximum of 5 squares, therefore a(4) = a(5) = 2.
%e A segment of length 3 can touch a maximum of 7 squares, therefore a(6) = a(7) = 3.
%t Table[If[n<=3,1,Ceiling[Sqrt[(n-3)^2/2+1]]],{n,70}] (* _Stefano Spezia_, Aug 03 2021 *)
%Y Cf. A346232.
%K nonn,easy
%O 1,4
%A _Luis Mendo_, Aug 02 2021
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