A236109 Triangle read by rows: another version of A048158, only here the representation of A004125 is symmetric, as in the representation of A024916 and A000203.

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, 0, 0, 0, 0, 0, 0, 0, 3

Offset

1,14

Comments

Row sums give A004125.

For more information see A236104, A237591, A237593, A237270.

Links

Table of n, a(n) for n=1..86.

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:
--------------------------------------------------
n     A000203  A024916            Diagram
--------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _
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

Cf. A000203, A004125, A024916, A048158, A196020, A235799, A236104, A236630, A236631, A237591, A237593, A237270.

Sequence in context: A281083 A230629 A027359 * A279279 A035447 A227188

Adjacent sequences: A236106 A236107 A236108 * A236110 A236111 A236112

Keyword

nonn,tabl

Author

Omar E. Pol, Jan 26 2014

Status

approved