Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd

622 T. Aaron Gulliver
2. Pentagonal Pyramids of Integers
A pentagonal pyramidal array of integers has a structure with 1 at the top,
2 to 6 on the second level, 7 to 18 on the third level, etc. An illustration of
the fourth level is give in Fig. 1. The number of elements on level i is a
Figure 1: The fourth level of the pentagonal pyramid of integers.
pentagonal number given by
1
i(3i − 1)
2
and the resulting integer sequence is
s i =1, 5, 12, 22, 35,...
The number of elements in the pyramid is then
n∑ 1
2 i(3i − 1) = 1 2 n2 (n +1), (3)
i=1
where n is the height of the pyramid. Starting from n = 1, we have
1, 6, 18, 40, 75,..., (4)
which is sequence A002411 [5], and appropriately called the pentagonal pyramidal
numbers.
A number of new and existing sequences can be obtained, depending on
the arrangement of numbers on a level. In this paper, we consider two different
arrangements. The first has the numbers increasing from one side of the level
to the other, while the other has numbers increasing in successive pentagons
on the level. For the top two levels, the arrangements are the same
1 ,
2
3 6
4 5