# generating function for interesting sequence

Andrew Plewe aplewe at sbcglobal.net
Fri Jul 8 23:01:04 CEST 2005

```I've stumbled on an interesting sequence (although for some reason it looks
familiar...) and I'm looking for a generating function a(n).  First, a few
terms of the sequence:

3, 5, 8, 7, 12, 15, 9, 16, 21, 24, 11, 20, 27, 32, 35, 13, 24, 33, 40, 45,
48

Which is a triangle read by anti-diagonals:

3   8   15  24  35  48
5   12  21  32  45
7   16  27  40
9   20  33
11  24
13

This sequence has the nice property that all the values in any given column
are divisible by the ordinal of that column, with the ordinal of the first
column = 1 (i.e. 8, 12, and 16 are all divisible by 2.  15, 21, 27 are all
divisible by 3, etc.).

Additively, the sequence looks like this:

3, 5, (5 + 3), 7, (7 + 5), (7 + 5 + 3), 9, (9 + 7), (9 + 7 + 5), (9 + 7 + 5
+ 3) . . .

and here is a graphical representation of how to generate the sequence [a(n)
= sum of terms in nth row]:

3
5
3 5
7
5 7
3 5 7
9
7 9
5 7 9
3 5 7 9
11
9 11
7 9 11
5 7 9 11
3 5 7 9 11
13
11 13
9 11 13
7 9 11 13
5 7 9 11 13
3 5 7 9 11 13

Several sequences already in the OEIS appear to be contained in this one;
A033428 = the main diagonal values (3, 12, 27, ...), other diagonals seem to
contain other sequences -- A045944 = (5, 16, 33, ...), A000567 [with the
exception of the first term] = (8, 21, 40, ...).

I'll submit the sequence to the OEIS later this evening.  Thanks!

-Andrew Plewe-

```