Download Sequences from Pentagonal Pyramids of Integers

International Mathematical Forum, 5, 2010, no. 13, 621 - 628
Sequences from Pentagonal Pyramids
of Integers
T. Aaron Gulliver
Department of Electrical and Computer Engineering
University of Victoria, P.O. Box 3055, STN CSC
Victoria, BC, V8W 3P6, Canada
[email protected]
Abstract
This paper presents a number of sequences based on integers arranged in a pentagonal pyramid structure. This approach provides a
simple derivation of some well known sequences. In addition, a number
of new integer sequences are obtained.
Mathematics Subject Classification: 11Y55
Keywords: integer arrays, integer sequences
1. Introduction
Previously, several well-known sequences (and many new sequences), were
derived from tetrahedral (three-sided) [2] and square (four-sided) [3] pyramids
of integers. For example, the number of elements in the square pyramid is
sn = 12 + 22 + 32 + 42 + 52 + . . . + n2 =
n
i=1
1
i2 = n(n + 1)(2n + 1),
6
(1)
where n is the height of the pyramid. Starting from n = 1, we have
1, 5, 14, 30, 55, . . . ,
(2)
which is sequence A000330 in the Encyclopedia of Integer Sequences maintained by Sloane [5], and appropriately called the square pyramidal numbers.
Sequences based on a pentagonal pyramid are given in the next section.
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T. Aaron Gulliver
2. Pentagonal Pyramids of Integers
A pentagonal pyramidal array of integers has a structure with 1 at the top,
2 to 6 on the second level, 7 to 18 on the third level, etc. An illustration of
the fourth level is give in Fig. 1. The number of elements on level i is a
Figure 1: The fourth level of the pentagonal pyramid of integers.
pentagonal number given by
1
i(3i − 1)
2
and the resulting integer sequence is
si = 1, 5, 12, 22, 35, . . .
The number of elements in the pyramid is then
n
1
i=1
1
i(3i − 1) = n2 (n + 1),
2
2
(3)
where n is the height of the pyramid. Starting from n = 1, we have
1, 6, 18, 40, 75, . . . ,
(4)
which is sequence A002411 [5], and appropriately called the pentagonal pyramidal numbers.
A number of new and existing sequences can be obtained, depending on
the arrangement of numbers on a level. In this paper, we consider two different
arrangements. The first has the numbers increasing from one side of the level
to the other, while the other has numbers increasing in successive pentagons
on the level. For the top two levels, the arrangements are the same
2
1 ,
3
6
4 5
623
Sequences from pentagonal pyramids of integers
For the third level, we have
7
7
8
17
11
8
14
9
18
10
and
16
11
9
10
12
18
13
12 13 15
.
17
14 15 16
In addition to (4), the following simple sequences are obtained from the integers
on the other corners of the first pyramid arrangement
1,
1,
1,
1,
2, 7, 19,
3, 9, 22,
4, 12, 28,
5, 15, 34,
41,
45,
55,
65,
...
...
.
...
...
The first of these is sequence A100119, the n-th centered n − 1-gonal number,
and is given by
1
sn = (n3 − 2n2 + n + 2).
2
The second is sequence A064808, the nth n + 1-gonal number, given by
1
sn = n(n2 − 2n + 3),
2
while the third is sequence A047732, the n-th n + 2-gonal number, given by
1
sn = n(n2 − n + 2).
2
The last sequence is A006003, given by
sn =
n(n2 + 1)
.
2
The corners of the second pyramid arrangement provide the following sequences
1,
1,
1,
1,
2, 7, 19, 41,
3, 12, 31, 63,
4, 14, 34, 67,
5, 16, 37, 71,
...
...
.
...
...
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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
1 3
(n + n2 − 6n + 6)
2
1 3
=
(n + n2 − 4n + 4)
2
1 3
(n + n2 − 2n + 2)
=
2
sn =
(5)
sn
(6)
sn
(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
1 3
(n − 2n2 + 2n + 1)
2
1
n(n3 − n2 − n + 3)
=
2
sn =
sn
(9)
(10)
respectively.
Now consider wedges in the first pyramid. The sum of the elements in the
leftmost wedge results in the sequence
1, 9, 57, 235, 720, . . . ,
with terms
1
sn = n(n + 1)(2n3 − 3n2 + 3n + 2).
8
Combining the two leftmost wedges gives sequence A101376
1, 14, 99, 424, 1325, . . . ,
with terms
sn =
=
n2
1
2
i=1 2 n(n − 1) + i
1 2
n (n3 − n2 + n + 1).
2
625
Sequences from pentagonal pyramids of integers
This sequence first appeared in [4] as one side of a cube of integers. In this
case the expression is
n n
2
sn =
j=1 n (i − 1) + j
i=1
n
1
2
2
=
i=1 2 n(2n i − 2n + n + 1)
= 12 n2 (n3 − n2 + n + 1)
The correspondence between the two shapes is not obvious. Adding the third
wedge gives the sum of the numbers on the pyramid level
1, 20, 150, 649, 2030, . . . ,
with terms
1
sn = n(3n − 1)(2n3 − n2 + n + 2).
8
Now, adding the terms in the above sequence gives the sum of the elements in
the entire pyramid
1, 21, 171, 820, 2850, . . . ,
with
sn =
=
n
1
3
i=1 8 i(3i − 1)(2i
1 2
n (n3 + n2 + 2).
8
− i2 + i + 2)
This is also equal to
sn =
=
12 n2 (n+1)
i=1
1 2
n
(n3
8
i
+ n2 + 2).
The partial sum of the pentagonal pyramidal numbers is sequence A001296
1, 7, 25, 65, 140, . . . ,
with
sn =
1
n(n + 1)(n + 2)(3n + 1)
24
which is 3n+1
C(n + 2, 3).
4
Returning to the second pyramid, one can take the sum of the corner
elements, i.e.
1, 2 + 4, 7 + 9 + 14, 19 + 21 + 26 + 34, . . . ,
(11)
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T. Aaron Gulliver
or
1, 6, 30, 100, 255, . . . ,
which is sequence A101375 [4], with
sn =
1
n(n
2
+ 1)(n2 − 2n + 2).
Taking the next set of corners gives
1, 2 + 5, 7 + 10 + 16, 19 + 22 + 28 + 37, . . .
(12)
or
1, 7, 33, 106, 265, . . . ,
which is sequence A100855 [4], with
sn =
1
n(n3
2
− n2 + n + 1).
Finally, taking the rightmost corners results in the sequence
1, 2 + 6, 7 + 10 + 16, 19 + 23 + 30 + 40, . . . ,
(13)
or
1, 8, 36, 112, 275, . . . ,
which is sequence A092365 [4], with
sn =
1 2
n (n2
2
− n + 2).
This last expression is also n2 [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)
n 1 3
2
sn =
i=1 2 i(i − i − i + 3)
= 12 n(n + 1)(4n3 + n2 − 11n + 26).
which gives the sequence
1, 6, 33, 127, 372, . . . ,
The sum of the elements in (11) is
n 1
2
sn =
i=1 2 i(i + 1)(i − 2i + 2)
1
n(n + 1)(n + 2)(12n2 − 21n + 29).
= 120
Sequences from pentagonal pyramids of integers
627
which gives the sequence
1, 7, 37, 137, 392, . . . ,
The sum of the elements in (12) is
n 1 3
2
sn =
i=1 2 i(i − i + i + 1)
1
= 120
n(n + 1)(12n3 + 3n2 + 7n + 38).
which gives the sequence
1, 8, 41, 147, 412, . . . ,
Finally, the sum of the elements in (13) is
n 1 2 2
sn =
i=1 2 i (i − i + 2)
1
= 40 n(n + 1)(4n3 + n2 + 9n + 6).
which gives the sequence
1, 9, 45, 157, 432, . . . ,
Returning to the first pyramid of integers, taking the wedges between the
rays, we have for the sum of the integers in the first wedge on the left
0, 0, 10, 23 + 24 + 25, 46 + 47 + 48 + 49 + 50 + 51, . . . ,
(14)
or
0, 0, 10, 72, 291, . . . ,
with
sn =
1
(n
8
− 1)(n − 2)(2n3 − 3n2 + 3n + 4).
The next wedge gives
0, 0, 13, 29 + 30 + 31, 56 + 57 + 58 + 59 + 60 + 61, . . . ,
or
0, 0, 13, 90, 351, . . . ,
with
sn =
1
(n
8
+ 1)(n − 1)(n − 2)(2n2 − 3n + 4).
(15)
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T. Aaron Gulliver
Finally, the last wedge gives
0, 0, 16, 35 + 36 + 37, 66 + 67 + 68 + 69 + 70 + 71, . . . ,
(16)
or
0, 0, 16, 108, 411, . . . ,
with
sn =
1
(n
8
− 1)(n − 2)(2n3 + n2 − n + 4).
References
[1] T.A. Gulliver, Sequences from Arrays of Integers, Int. Math. J. 1 323–332
(2002).
[2] T.A. Gulliver, Sequences from Integer Tetrahedrons, Int. Math. Forum, 1,
517–521 (2006).
[3] T.A. Gulliver, Sequences from Pyramids of Integers, Int. J. Pure and Applied Math. 36 161–165, (2007).
[4] T.A. Gulliver, Sequences from Cubes of Integers, Int. Math. J. 4, 439–445,
(2003). Correction Int. Math. Forum, vol. 1, no, 11, pp. 523-524.
[5] N.J.A. Sloane,
On-Line Encyclopedia of Integer
http://www.research.att.com/˜njas/sequences/index.html.
Received: June, 2009
Sequences,