Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Sequences</strong> <strong>from</strong> pentagonal pyramids <strong>of</strong> integers 623<br />
For the third level, we have<br />
7<br />
8 17<br />
11 14<br />
9 18<br />
10 16<br />
12 13 15<br />
and<br />
7<br />
8 11<br />
9 10<br />
12 18<br />
13 17<br />
14 15 16<br />
.<br />
In addition to (4), the following simple sequences are obtained <strong>from</strong> the integers<br />
on the other corners <strong>of</strong> the first pyramid arrangement<br />
1, 2, 7, 19, 41, ...<br />
1, 3, 9, 22, 45, ...<br />
1, 4, 12, 28, 55, ...<br />
1, 5, 15, 34, 65, ...<br />
The first <strong>of</strong> these is sequence A100119, the n-th centered n − 1-gonal number,<br />
and is given by<br />
s n = 1 2 (n3 − 2n 2 + n +2).<br />
The second is sequence A064808, the nth n + 1-gonal number, given by<br />
s n = 1 2 n(n2 − 2n +3),<br />
while the third is sequence A047732, the n-th n + 2-gonal number, given by<br />
s n = 1 2 n(n2 − n +2).<br />
The last sequence is A006003, given by<br />
s n = n(n2 +1)<br />
.<br />
2<br />
The corners <strong>of</strong> the second pyramid arrangement provide the following sequences<br />
1, 2, 7, 19, 41, ...<br />
1, 3, 12, 31, 63, ...<br />
1, 4, 14, 34, 67, ...<br />
1, 5, 16, 37, 71, ...<br />
.<br />
.
Change language