

A346693


Minimum integer length of a segment that touches the interior of n squares on a unit square grid.


1



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, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46, 47, 48
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


COMMENTS

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}.
The sequence is nondecreasing. 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.
There can be no more than 3 different consecutive terms.
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).


LINKS

Table of n, a(n) for n=1..70.


FORMULA

a(n) = 1 for n<=3; a(n) = ceiling(sqrt((n3)^2/2+1)) for n>= 4.


EXAMPLE

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.
A segment of length 2 can touch a maximum of 5 squares, therefore a(4) = a(5) = 2.
A segment of length 3 can touch a maximum of 7 squares, therefore a(6) = a(7) = 3.


MATHEMATICA

Table[If[n<=3, 1, Ceiling[Sqrt[(n3)^2/2+1]]], {n, 70}] (* Stefano Spezia, Aug 03 2021 *)


CROSSREFS

Cf. A346232.
Sequence in context: A283371 A116579 A156253 * A265436 A060151 A285902
Adjacent sequences: A346690 A346691 A346692 * A346694 A346695 A346696


KEYWORD

nonn,easy


AUTHOR

Luis Mendo, Aug 02 2021


STATUS

approved



