A344719: Minimum number of cells in an n X n grid that need to be painted so that each unpainted cell is orthogonally adjacent to exactly two painted cell. 
Example solutions for n <= 11. Painted cells are 'x' and unpainted cells are '.'. Later terms all have a distinct "diagonal" pattern.
By Dmitry Kamenetsky, 28/05/2021.

a(1)=1
x

a(2)=2
x.
.x

a(3)=5
.x.
xxx
.x.

a(4)=8
.xx.
x..x
x..x
.xx.

a(5)=11
.x.x.
xx..x
..x..
x..xx
.x.x.

a(6)=17
.x.xxx
x..x..
..x..x
xx..x.
x..xx.
x.x..x

a(7)=21, first example of the "diagonal" pattern.
.x.x..x
xx..xx.
..x..x.
x..x..x
.x..x..
.xx..xx
x..x.x.

a(8)=28
.x.xx.x.
x..x..xx
..x..x..
xx..x..x
x..x..xx
..x..x..
xx..x..x
.x.xx.x.

a(9)=35
.x.x..xx.
xx..xx..x
..x..x..x
x..x..xx.
.x..x..x.
.xx..x..x
x..x..x..
x..xx..xx
.xx..x.x.

a(10)=42
.x.xx.x..x
xx..x..xx.
..x..x..x.
x..x..x..x
xx..x..x..
..x..x..xx
x..x..x..x
.x..x..x..
.xx..x..xx
x..x.xx.x.

a(11)=51
.x.xx.xx.x.
xx..x..x..x
..x..x..x..
x..x..x..xx
xx..x..x..x
..x..x..x..
x..x..x..xx
xx..x..x..x
..x..x..x..
x..x..x..xx
.x.xx.xx.x.