Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd

624 T. Aaron Gulliver
The first sequence is the same as that above, but the others do not appear in
the database and so are new. These sequences are given by
s n = 1 2 (n3 + n 2 − 6n + 6) (5)
s n = 1 2 (n3 + n 2 − 4n + 4) (6)
s n = 1 2 (n3 + n 2 − 2n + 2) (7)
respectively. All subsequent sequences in this paper are also new, unless otherwise
noted.
The sum of the elements on the left bottom row of the pyramids gives the
sequences
1, 5, 24, 82, 215, ...
1, 5, 27, 94, 245, ... (8)
with
s n = 1 2 (n3 − 2n 2 +2n + 1) (9)
s n = 1 2 n(n3 − n 2 − n + 3) (10)
respectively.
Now consider wedges in the first pyramid. The sum of the elements in the
leftmost wedge results in the sequence
with terms
1, 9, 57, 235, 720, ... ,
s n = 1 8 n(n + 1)(2n3 − 3n 2 +3n +2).
Combining the two leftmost wedges gives sequence A101376
with terms
1, 14, 99, 424, 1325, ... ,
s n = ∑ n 2
i=1
1
2 n(n − 1)2 + i
= 1 2 n2 (n 3 − n 2 + n +1).