OFFSET
1,1
COMMENTS
Row n is a palindromic composition of sigma(4n-1).
Row n is also the row 4n-1 of A237270.
Row n has length A237271(4n-1).
Row sums give A239053.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the third quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-2), see A239932.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016
EXAMPLE
The irregular triangle begins:
2, 2;
4, 4;
6, 6;
8, 8, 8;
10, 10;
12, 12;
14, 6, 6, 14;
16, 16;
18, 12, 18;
20, 8, 8, 20;
22, 22;
24, 24;
26, 10, 10, 26;
28, 8, 8, 28;
30, 30;
32, 12, 16, 12, 32;
...
Illustration of initial terms in the third quadrant of the spiral described in A239660:
. _ _ _ _ _ _ _ _
. | | | | | | | | | | | | | | | |
. | | | | | | | | | | | | | | |_|_ _
. | | | | | | | | | | | | | | 2 |_ _|
. | | | | | | | | | | | | |_|_ 2
. | | | | | | | | | | | | 4 |_
. | | | | | | | | | | |_|_ _ |_ _ _ _
. | | | | | | | | | | 6 |_ |_ _ _ _|
. | | | | | | | | |_|_ _ _ |_ 4
. | | | | | | | | 8 | |_ _ |
. | | | | | | |_|_ _ _ |_ | |_ _ _ _ _ _
. | | | | | | 10 | |_ |_ |_ _ _ _ _ _|
. | | | | |_|_ _ _ _ |_ _ 8 |_ _| 6
. | | | | 12 | |_ |
. | | |_|_ _ _ _ _ |_ _ | |_ _ _ _ _ _ _ _
. | | 14 | | |_ |_ _ |_ _ _ _ _ _ _ _|
. |_|_ _ _ _ _ | |_ _ |_ | 8
. 16 | |_ _ | | |
. | |_|_ |_ _ |_ _ _ _ _ _ _ _ _ _
. |_ _ 6 |_ _ | |_ _ _ _ _ _ _ _ _ _|
. | |_ | | 10
. |_ 6 | |_ _ |
. |_ |_ _ _| |_ _ _ _ _ _ _ _ _ _ _ _
. |_ _ | |_ _ _ _ _ _ _ _ _ _ _ _|
. | | 12
. |_ _ _ |
. | |_ _ _ _ _ _ _ _ _ _ _ _ _ _
. | |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
. | 14
. |
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
. 16
.
For n = 7 we have that 4*7-1 = 27 and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] therefore between both Dyck paths there are four regions (or parts) of sizes [14, 6, 6, 14], so row 7 is [14, 6, 6, 14].
The sum of divisors of 27 is 1 + 3 + 9 + 27 = A000203(27) = 40. On the other hand the sum of the parts of the symmetric representation of sigma(27) is 14 + 6 + 6 + 14 = 40, equaling the sum of divisors of 27.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Mar 29 2014
STATUS
approved