Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd
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<strong>Sequences</strong> <strong>from</strong> pentagonal pyramids <strong>of</strong> integers 627<br />
which gives the sequence<br />
The sum <strong>of</strong> the elements in (12) is<br />
which gives the sequence<br />
1, 7, 37, 137, 392, ... ,<br />
s n = ∑ n 1<br />
i=1 2 i(i3 − i 2 + i +1)<br />
= 1 n(n + 120 1)(12n3 +3n 2 +7n + 38).<br />
1, 8, 41, 147, 412, ... ,<br />
Finally, the sum <strong>of</strong> the elements in (13) is<br />
which gives the sequence<br />
s n = ∑ n 1<br />
i=1 2 i2 (i 2 − i +2)<br />
= 1 n(n + 40 1)(4n3 + n 2 +9n +6).<br />
1, 9, 45, 157, 432, ... ,<br />
Returning to the first pyramid <strong>of</strong> integers, taking the wedges between the<br />
rays, we have for the sum <strong>of</strong> the integers in the first wedge on the left<br />
or<br />
with<br />
0, 0, 10, 23 + 24 + 25, 46 + 47 + 48 + 49 + 50 + 51, ... , (14)<br />
The next wedge gives<br />
or<br />
with<br />
0, 0, 10, 72, 291, ... ,<br />
s n = 1 8 (n − 1)(n − 2)(2n3 − 3n 2 +3n +4).<br />
0, 0, 13, 29 + 30 + 31, 56 + 57 + 58 + 59 + 60 + 61, ... , (15)<br />
0, 0, 13, 90, 351, ... ,<br />
s n = 1 8 (n + 1)(n − 1)(n − 2)(2n2 − 3n +4).
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