The following diagrams show a square dissection into regions that are the symmetric representation of A000203, A004125, A153485 and this sequence.
In order to construct every diagram we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593.
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n).
At stage 3 we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals a(n).
At stage 4 we draw a copy of the symmetric representation of A004125(n) rotated 180 degrees such that one of its vertices is the point (0,0). The area of the region (or regions) that is above of this region and below the staircase equals A153485(n).
Illustration for n = 1..6:
. _ _ _ _ _ _
. _ _ _ _ _ |_ _ _ |_ R|
. _ _ _ _ R |_ _S_| R| | |_T | S |_|
. _ _ _ R |_ _ |_| | |_ |_ _| | |_|_ _ |
. _ _ |_S_|_| | |_|_S | |_U_|_T | | |_ U |_T | |
. _ S |_ S| U|_|_|S| |_ U|_| | | | |_|S| | |_ |_| |
. |_| |_|_| |_|_|_| |_|_ _|_| |_V_|_U_|_| |_V_|_ _ _|_|
. U V U V
.
n: 1 2 3 4 5 6
T: a(n) 0 0 1 2 5 6
.
Illustration for n = 7..9:
. _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ |_ _ _S_ _| |
. _ _ _ _ _ _ _ |_ _ _ _ | | | |_ |_ _ R |
. |_ _S_ _| | | |_ | |_ R | | |_ |_ S| |
. | |_ |_ R | | |_ |_S |_ _| | |_ T |_|_ _|
. | |_ T |_ _| | |_T |_ _ | |_ _ |_ | |
. |_ _ |_ | | |_ _ U |_ | | | | U |_ | |
. | |_U |_ |S| | |_ |_ | | | |_ _ |_ |S|
. | V | |_| | | V | |_| | | V | |_| |
. |_ _ _|_ _ _|_| |_ _ _|_ _ _ _|_| |_ _ _ _|_ _ _ _|_|
.
n: 7 8 9
T: a(n) 12 13 20
.
Illustration for n = 10..12:
. _ _ _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ _ _ _ |_ _ _ _ _ _ | |
. _ _ _ _ _ _ _ _ _ _ |_ _ _S_ _ _| | | |_ | |_ _ R |
. |_ _ _S_ _ | | | |_ | R | | |_ | |_ |
. | |_ | |_ R | | |_ |_ | | |_ |_ S | |
. | |_ |_ _|_ | | |_ |_ | | |_ |_ |_ _|
. | |_ | |_ _| | |_ T |_ _ _| | |_ T |_ _ _ |
. | |_ T |_ _ | |_ _ _ |_ | | |_ _ |_ | |
. |_ _ |_ | | | |_ U |_ | | | | U |_ | |
. | |_ U |_ |S| | |_ |_ |S| | |_ |_ | |
. | |_ |_ | | | | |_ | | | |_ _ |_ | |
. | V | |_| | | V | |_| | | V | |_| |
. |_ _ _ _|_ _ _ _ _|_| |_ _ _ _ _|_ _ _ _ _|_| |_ _ _ _ _|_ _ _ _ _ _|_|
.
n: 10 11 12
T: a(n) 24 32 33
.
Note that in the diagrams the symmetric representation of a(n) is the same as the symmetric representation of A153485(n-1) rotated 180 degrees.
The original examples (dated Jun 24 2014) were only the diagrams for n = 11 and n = 12. (End)
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