%e A168368 Following is a graphic depiction of the stable connected piles of bricks 
             for n = 0 to 9 ordered by increasing height (all piles of a given 
             height within curly braces) and each variant of a given pattern within 
             square brackets, where: C(k, i) is k choose i (binomial coefficient), 
             F_n is n_th Fibonacci number [F_n = Sum_{k+i = n-1, i <= k} C(k, 
             i)].  Also, the piles of heights 1 and 2 are grouped within parentheses
             (since they give the n_th Fibonacci number.)
%e A168368 For n = 0, the following 0 [F_0] pile:
             ( { } )
%e A168368 For n = 1, the following 1 [F_1 = C(0, 0) = 1] pile:
             ( { |=| } )
%e A168368 For n = 2, the following 1 [F_2 = C(1, 0) = 1] pile:
             ( { |=|=| } )
%e A168368 For n = 3, the following 2 [F_3 = C(2, 0) + C(1, 1) = 2] piles:
             ( { ....... } & { .|=|. } )
             ( { |=|=|=| } & { |=|=| } )
%e A168368 For n = 4, the following 4 [F_4 + 1 = (C(3, 0) + C(2, 1)) + 1 = 3 + 1] 
             piles (where the brick on the third level is necessary for stability):
             ( { ......... } & { ....... & ....... } ) & { .|=|. }
             ( { ......... } & { .|=|... & ...|=|. } ) & { |=|=| }
             ( { |=|=|=|=| } & { |=|=|=| & |=|=|=| } ) & { .|=|. }
%e A168368 For n = 5, the following 7 [F_5 + 2 = (C(4, 0) + C(3, 1) + C(2, 2)) + 
             2 = 5 + 2] piles:
             ( { ........... } &
             ( { |=|=|=|=|=| } &
             and
             { [ .|=|..... & ...|=|... & .....|=|. ] & .|=|=|. } ) &
             { [ |=|=|=|=| & |=|=|=|=| & |=|=|=|=| ] & |=|=|=| } ) &
             and (where the brick on the third level is necessary for stability)
             { .|=|.. & ..|=|. }
             { |=|=|. & .|=|=| }
             { .|=|=| & |=|=|. }
%e A168368 For n = 6, the following 12 [F_6 + 4 = (C(5, 0) + C(4, 1) + C(3, 2)) 
             + 4 = 8 + 4] piles:
             ( { ............. } &
             ( { |=|=|=|=|=|=| } &
             and
             { [ .|=|....... & (C(4,1) = 4 ways) & .......|=|. ] &
             { [ |=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=| ] &
             and
             [ .|=|=|... & .|=|.|=|. & ...|=|=|. ] } ) &
             [ |=|=|=|=| & |=|=|=|=| & |=|=|=|=| ] } ) &
             and (where 3rd level brick is necessary for stability of first and
             third piles, samely for 4th level brick of 4th pile)
             { ........ & ....... & ........ } & { .|=|. }
             { .|=|.... & ..|=|.. & ....|=|. } & { |=|=| }
             { |=|=|... & .|=|=|. & ...|=|=| } & { .|=|. }
             { .|=|=|=| & |=|=|=| & |=|=|=|. } & { |=|=| }
%e A168368 For n = 7, the following 21 [F_7 + 8 = (C(6, 0) + C(5, 1) + C(4, 2) + 
             C(3, 3)) + 8 = 13 + 8] piles:
             ( { ............... } &
             ( { |=|=|=|=|=|=|=| } &
             and
             { [ .|=|......... & (C(5,1) = 5 ways) & .........|=|. ] &
             { [ |=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=| ] &
             and
             [ .|=|=|..... & (C(4,2) = 6 ways) & .....|=|=|. ] & .|=|=|=|. } ) &
             [ |=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=| ] & |=|=|=|=| } ) &
             and (where the brick on the third level is necessary for stability
             of first and last piles)
             { [ .|=|...... & ...|=|.... & ....|=|... & ......|=|. ] &
             { [ |=|=|..... & ..|=|=|... & ...|=|=|.. & .....|=|=| ] &
             { [ .|=|=|=|=| & .|=|=|=|=| & |=|=|=|=|. & |=|=|=|=|. ] &
             and (where all the bricks on the third level are necessary for stability)
             [ .|=|.... & ....|=|. ] & .|=|=|. } &
             [ |=|=|=|. & .|=|=|=| ] & |=|=|=| } &
             [ .|=|=|=| & |=|=|=|. ] & .|=|=|. } &
             and (where the bricks on the third and fifth levels are necessary for 
             stability)
             { .|=|. }
             { |=|=| }
             { .|=|. }
             { |=|=| }
             { .|=|. }
%e A168368 For n = 8, the following 40 [F_8 + 19 = (C(7, 0) + C(6, 1) + C(5, 2) + 
             C(4, 3)) + 19 = 21 + 19] piles:
             ( { ................. } &
             ( { |=|=|=|=|=|=|=|=| } &
             and
             { [ .|=|........... & (C(6,1) = 6 ways) & ...........|=|. ] &
             { [ |=|=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=|=| ] &
             and
             [ .|=|=|....... & (C(5,2)= 10 ways) & .......|=|=|. ] &
             [ |=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=| ] &
             and
             [ .|=|=|=|... & (C(4,3) = 4 ways) & ...|=|=|=|. ] } ) &
             [ |=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=| ] } ) &
             and
             { [ .|=|........ & . . . . . & ........|=|. ] &
             { [ |=|=|....... & (5  ways) & .......|=|=| ] &
             { [ .|=|=|=|=|=| & . . . . . & |=|=|=|=|=|. ] &
             and (where either |1| or |2| is a brick |=|, giving 6 ways)
             [ .|=|...... & ..|1|2|.. & ......|=|. ] &
             [ |=|=|1|2|. & .|=|=|=|. & .|2|1|=|=| ] &
             [ .|=|=|=|=| & |=|=|=|=| & |=|=|=|=|. ] &
             and (this one not contiguous on first row)
             .|=|.|=|. } &
             |=|=|=|=| } &
             .|=|.|=|. } &
             and (where either |1| or |2| is a brick |=|, giving 5 ways)
             { [ .|=|..... & ...|=|... & .....|=|. ] } &
             { [ |=|=|.... & ..|=|=|.. & ....|=|=| ] } &
             { [ .|=|2|... & ...|=|... & ...|2|=|. ] } &
             { [ |=|=|=|1| & |=|=|=|=| & |1|=|=|=| ] } &
             and
             { [ .|=|.. & ..|=|. ] }
             { [ |=|=|. & .|=|=| ] }
             { [ .|=|.. & ..|=|. ] }
             { | |=|=|. & .|=|=| ] }
             { [ .|=|=| & |=|=|. ] }
%e A168368 For n = 9, the following 67 [F_9 + 33 = (C(8, 0) + C(7, 1) + C(6, 2) + 
             C(5, 3) + C(4, 4)) + 33 = 34 + 33] piles:
             ( { ................... } &
             ( { |=|=|=|=|=|=|=|=|=| } &
             and
             { [ .|=|............. & (C(7,1) = 7 ways) & .............|=|. ] &
             { [ |=|=|=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=|=|=| ] &
             and
             [ .|=|=|......... & (C(6,2)= 15 ways) & .........|=|=|. ] &
             [ |=|=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=|=| ] &
             and
             [ .|=|=|=|..... & (C(5,3)= 10 ways) & .....|=|=|=|. ] &
             [ |=|=|=|=|=|=| & . . . . . . . . . & |=|=|=|=|=|=| ] &
             and
             .|=|=|=|=|. } ) &
             |=|=|=|=|=| } ) &
             and
             { [ .|=|.......... & . . . . . & ..........|=|. ] &
             { [ |=|=|......... & (6  ways) & .........|=|=| ] &
             { [ .|=|=|=|=|=|=| & . . . . . & |=|=|=|=|=|=|. ] &
             and (where either |1|, |2| or |3| is a brick |=|, giving 6 ways)
             [ .|=|........ & ..|1|2|.... & ..|=|...... &
             [ |=|=|1|2|3|. & .|=|=|=|... & .|=|=|.|=|. &
             [ .|=|=|=|=|=| & |=|=|=|=|=| & |=|=|=|=|=| &
             and (where either |1|, |2| or |3| is a brick |=|, giving 6 ways)
             ......|=|.. & ....|2|1|.. & ........|=|. ] } &
             .|=|.|=|=|. & ...|=|=|=|. & .|3|2|1|=|=| ] } &
             |=|=|=|=|=| & |=|=|=|=|=| & |=|=|=|=|=|. ] } &
             and (where either |1|, |2|, |3| or |4| is a brick |=|, giving 10 ways)
             { [ .|=|....... & .....|=|..... & .......|=|. ] } &
             { [ |=|=|...... & ....|=|=|.... & ......|=|=| ] } &
             { [ .|=|2|3|... & ...|3|=|4|... & ...|3|2|=|. ] } &
             { [ |=|=|=|=|1| & |1|=|=|=|=|2| & |1|=|=|=|=| ] } &
             and
             { [ .|=|.... & ..|=|.. & ....|=|. ] & ..|=|.. } &
             { [ |=|=|... & .|=|=|. & ...|=|=| ] & .|=|=|. } &
             { [ .|=|.... & ..|=|.. & ....|=|. ] & |=|=|=| } &
             { [ |=|=|... & .|=|=|. & ...|=|=| ] & .|=|=|. } &
             { [ .|=|=|=| & |=|=|=| & |=|=|=|. ] & ..|=|.. } &
             and
             { .|=|. }
             { |=|=| }
             { .|=|. }
             { |=|=| }
             { .|=|. }
             { |=|=| }
%e A168368 I think I included all the stable connected piles of bricks for n=6, 
             n=7, n=8 and n=9, please do check!