Sequences from Pentagonal Pyramids of Integers - HIKARI Ltd

626 T. Aaron Gulliver
or
which is sequence A101375 [4], with
Taking the next set of corners gives
or
1, 6, 30, 100, 255, ... ,
s n = 1 2 n(n + 1)(n2 − 2n +2).
1, 2+5, 7 + 10 + 16, 19 + 22 + 28 + 37,... (12)
which is sequence A100855 [4], with
1, 7, 33, 106, 265, ... ,
s n = 1 2 n(n3 − n 2 + n +1).
Finally, taking the rightmost corners results in the sequence
or
1, 2+6, 7 + 10 + 16, 19 + 23 + 30 + 40, ... , (13)
which is sequence A092365 [4], with
1, 8, 36, 112, 275, ... ,
s n = 1 2 n2 (n 2 − n +2).
This last expression is also n 2 [C(n, 2) + 1]. One can also take the sums of the
elements in the rays of this pyramid. Considering the sum of the elements in
the second line in (8)
which gives the sequence
The sum of the elements in (11) is
s n = ∑ n 1
i=1 2 i(i3 − i 2 − i +3)
= 1n(n + 2 1)(4n3 + n 2 − 11n + 26).
1, 6, 33, 127, 372, ... ,
s n = ∑ n 1
i=1
i(i + 2 1)(i2 − 2i +2)
= 1 n(n + 1)(n + 120 2)(12n2 − 21n + 29).