Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd

Sequences from pentagonal pyramids of integers 625
This sequence first appeared in [4] as one side of a cube of integers. In this
case the expression is
s n = ∑ n
∑ n
i=1 j=1 n2 (i − 1) + j
= ∑ n 1
i=1 2 n(2n2 i − 2n 2 + n +1)
= 1 2 n2 (n 3 − n 2 + n +1)
The correspondence between the two shapes is not obvious. Adding the third
wedge gives the sum of the numbers on the pyramid level
with terms
1, 20, 150, 649, 2030, ... ,
s n = 1 8 n(3n − 1)(2n3 − n 2 + n +2).
Now, adding the terms in the above sequence gives the sum of the elements in
the entire pyramid
with
This is also equal to
with
1, 21, 171, 820, 2850, ... ,
s n = ∑ n 1
i=1
i(3i − 8 1)(2i3 − i 2 + i +2)
= 1 8 n2 (n 3 + n 2 +2).
s n = ∑ 1
2 n2 (n+1)
i=1 i
= 1 8 n2 (n 3 + n 2 +2).
The partial sum of the pentagonal pyramidal numbers is sequence A001296
1, 7, 25, 65, 140, ... ,
s n = 1 n(n + 1)(n + 2)(3n +1)
24
which is 3n+1 C(n +2, 3).
4
Returning to the second pyramid, one can take the sum of the corner
elements, i.e.
1, 2+4, 7+9+14, 19 + 21 + 26 + 34, ... , (11)