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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International Mathematical Forum, 5, 2010, no. 13, 621 - 628<br />
<strong>Sequences</strong> <strong>from</strong> <strong>Pentagonal</strong> <strong>Pyramids</strong><br />
<strong>of</strong> <strong>Integers</strong><br />
T. Aaron Gulliver<br />
Department <strong>of</strong> Electrical and Computer Engineering<br />
University <strong>of</strong> Victoria, P.O. Box 3055, STN CSC<br />
Victoria, BC, V8W 3P6, Canada<br />
agullive@ece.uvic.ca<br />
Abstract<br />
This paper presents a number <strong>of</strong> sequences based on integers arranged<br />
in a pentagonal pyramid structure. This approach provides a<br />
simple derivation <strong>of</strong> some well known sequences. In addition, a number<br />
<strong>of</strong> new integer sequences are obtained.<br />
Mathematics Subject Classification: 11Y55<br />
Keywords: integer arrays, integer sequences<br />
1. Introduction<br />
Previously, several well-known sequences (and many new sequences), were<br />
derived <strong>from</strong> tetrahedral (three-sided) [2] and square (four-sided) [3] pyramids<br />
<strong>of</strong> integers. For example, the number <strong>of</strong> elements in the square pyramid is<br />
s n =1 2 +2 2 +3 2 +4 2 +5 2 + ...+ n 2 =<br />
n∑<br />
i=1<br />
i 2 = 1 n(n + 1)(2n +1), (1)<br />
6<br />
where n is the height <strong>of</strong> the pyramid. Starting <strong>from</strong> n = 1, we have<br />
1, 5, 14, 30, 55,..., (2)<br />
which is sequence A000330 in the Encyclopedia <strong>of</strong> Integer <strong>Sequences</strong> maintained<br />
by Sloane [5], and appropriately called the square pyramidal numbers.<br />
<strong>Sequences</strong> based on a pentagonal pyramid are given in the next section.
622 T. Aaron Gulliver<br />
2. <strong>Pentagonal</strong> <strong>Pyramids</strong> <strong>of</strong> <strong>Integers</strong><br />
A pentagonal pyramidal array <strong>of</strong> integers has a structure with 1 at the top,<br />
2 to 6 on the second level, 7 to 18 on the third level, etc. An illustration <strong>of</strong><br />
the fourth level is give in Fig. 1. The number <strong>of</strong> elements on level i is a<br />
Figure 1: The fourth level <strong>of</strong> the pentagonal pyramid <strong>of</strong> integers.<br />
pentagonal number given by<br />
1<br />
i(3i − 1)<br />
2<br />
and the resulting integer sequence is<br />
s i =1, 5, 12, 22, 35,...<br />
The number <strong>of</strong> elements in the pyramid is then<br />
n∑ 1<br />
2 i(3i − 1) = 1 2 n2 (n +1), (3)<br />
i=1<br />
where n is the height <strong>of</strong> the pyramid. Starting <strong>from</strong> n = 1, we have<br />
1, 6, 18, 40, 75,..., (4)<br />
which is sequence A002411 [5], and appropriately called the pentagonal pyramidal<br />
numbers.<br />
A number <strong>of</strong> new and existing sequences can be obtained, depending on<br />
the arrangement <strong>of</strong> numbers on a level. In this paper, we consider two different<br />
arrangements. The first has the numbers increasing <strong>from</strong> one side <strong>of</strong> the level<br />
to the other, while the other has numbers increasing in successive pentagons<br />
on the level. For the top two levels, the arrangements are the same<br />
1 ,<br />
2<br />
3 6<br />
4 5
<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 />
.
624 T. Aaron Gulliver<br />
The first sequence is the same as that above, but the others do not appear in<br />
the database and so are new. These sequences are given by<br />
s n = 1 2 (n3 + n 2 − 6n + 6) (5)<br />
s n = 1 2 (n3 + n 2 − 4n + 4) (6)<br />
s n = 1 2 (n3 + n 2 − 2n + 2) (7)<br />
respectively. All subsequent sequences in this paper are also new, unless otherwise<br />
noted.<br />
The sum <strong>of</strong> the elements on the left bottom row <strong>of</strong> the pyramids gives the<br />
sequences<br />
1, 5, 24, 82, 215, ...<br />
1, 5, 27, 94, 245, ... (8)<br />
with<br />
s n = 1 2 (n3 − 2n 2 +2n + 1) (9)<br />
s n = 1 2 n(n3 − n 2 − n + 3) (10)<br />
respectively.<br />
Now consider wedges in the first pyramid. The sum <strong>of</strong> the elements in the<br />
leftmost wedge results in the sequence<br />
with terms<br />
1, 9, 57, 235, 720, ... ,<br />
s n = 1 8 n(n + 1)(2n3 − 3n 2 +3n +2).<br />
Combining the two leftmost wedges gives sequence A101376<br />
with terms<br />
1, 14, 99, 424, 1325, ... ,<br />
s n = ∑ n 2<br />
i=1<br />
1<br />
2 n(n − 1)2 + i<br />
= 1 2 n2 (n 3 − n 2 + n +1).
<strong>Sequences</strong> <strong>from</strong> pentagonal pyramids <strong>of</strong> integers 625<br />
This sequence first appeared in [4] as one side <strong>of</strong> a cube <strong>of</strong> integers. In this<br />
case the expression is<br />
s n = ∑ n<br />
∑ n<br />
i=1 j=1 n2 (i − 1) + j<br />
= ∑ n 1<br />
i=1 2 n(2n2 i − 2n 2 + n +1)<br />
= 1 2 n2 (n 3 − n 2 + n +1)<br />
The correspondence between the two shapes is not obvious. Adding the third<br />
wedge gives the sum <strong>of</strong> the numbers on the pyramid level<br />
with terms<br />
1, 20, 150, 649, 2030, ... ,<br />
s n = 1 8 n(3n − 1)(2n3 − n 2 + n +2).<br />
Now, adding the terms in the above sequence gives the sum <strong>of</strong> the elements in<br />
the entire pyramid<br />
with<br />
This is also equal to<br />
with<br />
1, 21, 171, 820, 2850, ... ,<br />
s n = ∑ n 1<br />
i=1<br />
i(3i − 8 1)(2i3 − i 2 + i +2)<br />
= 1 8 n2 (n 3 + n 2 +2).<br />
s n = ∑ 1<br />
2 n2 (n+1)<br />
i=1 i<br />
= 1 8 n2 (n 3 + n 2 +2).<br />
The partial sum <strong>of</strong> the pentagonal pyramidal numbers is sequence A001296<br />
1, 7, 25, 65, 140, ... ,<br />
s n = 1 n(n + 1)(n + 2)(3n +1)<br />
24<br />
which is 3n+1 C(n +2, 3).<br />
4<br />
Returning to the second pyramid, one can take the sum <strong>of</strong> the corner<br />
elements, i.e.<br />
1, 2+4, 7+9+14, 19 + 21 + 26 + 34, ... , (11)
626 T. Aaron Gulliver<br />
or<br />
which is sequence A101375 [4], with<br />
Taking the next set <strong>of</strong> corners gives<br />
or<br />
1, 6, 30, 100, 255, ... ,<br />
s n = 1 2 n(n + 1)(n2 − 2n +2).<br />
1, 2+5, 7 + 10 + 16, 19 + 22 + 28 + 37,... (12)<br />
which is sequence A100855 [4], with<br />
1, 7, 33, 106, 265, ... ,<br />
s n = 1 2 n(n3 − n 2 + n +1).<br />
Finally, taking the rightmost corners results in the sequence<br />
or<br />
1, 2+6, 7 + 10 + 16, 19 + 23 + 30 + 40, ... , (13)<br />
which is sequence A092365 [4], with<br />
1, 8, 36, 112, 275, ... ,<br />
s n = 1 2 n2 (n 2 − n +2).<br />
This last expression is also n 2 [C(n, 2) + 1]. One can also take the sums <strong>of</strong> the<br />
elements in the rays <strong>of</strong> this pyramid. Considering the sum <strong>of</strong> the elements in<br />
the second line in (8)<br />
which gives the sequence<br />
The sum <strong>of</strong> the elements in (11) is<br />
s n = ∑ n 1<br />
i=1 2 i(i3 − i 2 − i +3)<br />
= 1n(n + 2 1)(4n3 + n 2 − 11n + 26).<br />
1, 6, 33, 127, 372, ... ,<br />
s n = ∑ n 1<br />
i=1<br />
i(i + 2 1)(i2 − 2i +2)<br />
= 1 n(n + 1)(n + 120 2)(12n2 − 21n + 29).
<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).
628 T. Aaron Gulliver<br />
Finally, the last wedge gives<br />
0, 0, 16, 35 + 36 + 37, 66 + 67 + 68 + 69 + 70 + 71, ... , (16)<br />
or<br />
0, 0, 16, 108, 411, ... ,<br />
with<br />
s n = 1(n − 1)(n − 8 2)(2n3 + n 2 − n +4).<br />
References<br />
[1] T.A. Gulliver, <strong>Sequences</strong> <strong>from</strong> Arrays <strong>of</strong> <strong>Integers</strong>, Int. Math. J. 1 323–332<br />
(2002).<br />
[2] T.A. Gulliver, <strong>Sequences</strong> <strong>from</strong> Integer Tetrahedrons, Int. Math. Forum, 1,<br />
517–521 (2006).<br />
[3] T.A. Gulliver, <strong>Sequences</strong> <strong>from</strong> <strong>Pyramids</strong> <strong>of</strong> <strong>Integers</strong>, Int. J. Pure and Applied<br />
Math. 36 161–165, (2007).<br />
[4] T.A. Gulliver, <strong>Sequences</strong> <strong>from</strong> Cubes <strong>of</strong> <strong>Integers</strong>, Int. Math. J. 4, 439–445,<br />
(2003). Correction Int. Math. Forum, vol. 1, no, 11, pp. 523-524.<br />
[5] N.J.A. Sloane, On-Line Encyclopedia <strong>of</strong> Integer <strong>Sequences</strong>,<br />
http://www.research.att.com/˜njas/sequences/index.html.<br />
Received: June, 2009