OFFSET
1,14
EXAMPLE
Triangle begins:
0;
0, 0;
0, 0, 1;
0, 0, 0, 1;
0, 0, 0, 2, 2;
0, 0, 0, 0, 1, 2;
0, 0, 0, 0, 2, 3, 3;
0, 0, 0, 0, 0, 2, 3, 3;
0, 0, 0, 0, 0, 2, 2, 4, 4;
0, 0, 0, 0, 0, 0, 2, 3, 4, 4;
0, 0, 0, 0, 0, 0, 3, 4, 5, 5, 5;
0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 5;
...
For the symmetric representation of A000203, A024916, A004125 in the fourth quadrant using a diagram which arises from the sequence A236104 see below:
--------------------------------------------------
--------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _
1 1 1 |_| | | | | | | | | | | |
2 3 4 |_ _|_| | | | | | | | | |
3 4 8 |_ _| _|_| | | | | | | |
4 7 15 |_ _ _| _|_| | | | | |
5 6 21 |_ _ _| _| _ _|_| | | |
6 12 33 |_ _ _ _| _| | _ _|_| |
7 8 41 |_ _ _ _| |_ _|_| _ _|
8 15 56 |_ _ _ _ _| _| |* *
9 13 69 |_ _ _ _ _| | _|* *
10 18 87 |_ _ _ _ _ _| _ _|* * *
11 12 99 |_ _ _ _ _ _| |* * * * *
12 28 127 |_ _ _ _ _ _ _|* * * * *
.
The 12th row is ........ 0,0,0,0,0,0,0,2,2,3,5,5
.
The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n). It appears that the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n). Example: for n = 12 the 12th row of triangle is 144, 25, 9, 1, hence the alternating sums is 144 - 25 + 9 - 1 = 127. On the other hand we have that A000290(12) - A004125(12) = 144 - 17 = A024916(12) = 127, equaling the total number of cells in the diagram after 12 stages. The number of cells in the 12th set of symmetric regions of the diagram is sigma(12) = A000203(12) = 28. Note that in this case there is only one region. The number of "*"'s is A004125(12) = 17.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jan 26 2014
STATUS
approved