Download Sequences from Hexagonal Pyramid of Integers

International Mathematical Forum, Vol. 6, 2011, no. 17, 821 - 827
Sequences from Hexagonal Pyramid 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 hexagonal 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], square (four-sided) [3], and pentagonal (five-sided) [5] pyramids of integers. For example, the number of elements
in the square pyramid is
2
2
2
2
2
2
sn = 1 + 2 + 3 + 4 + 5 + . . . + n =
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 [6], and appropriately called the square pyramidal numbers.
Sequences based on a hexagonal pyramid are given in the next section.
822
T. Aaron Gulliver
2. Hexagonal Pyramids of Integers
A hexagonal pyramidal array of integers has a structure with 1 at the top,
2 to 7 on the second level, 7 to 22 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 hexagonal pyramid of integers.
hexagonal number given by
i(2i − 1)
and the resulting integer sequence is
si = 1, 6, 15, 28, 45, . . .
The number of elements in the pyramid is then
n
i=1
1
i(2i − 1) = n(n + 1)(4n − 1),
6
(3)
where n is the height of the pyramid. Starting from n = 1, we have
1, 7, 22, 50, 95, . . .
(4)
which is sequence A002412 in [6], and are called the hexagonal pyramidal
numbers.
A number of new sequences can be obtained from this structure, depending
on the arrangement of numbers on a level. In this paper, we consider the
following arrangement. For the top two levels, this is
2
1 ,
3
4
7
6
5
823
Sequences from hexagonal pyramid of integers
For the third level, we have
8
9
13
14 10
12 22
15
11
21 .
16
20
17
19
18
In addition to (4), the following simple sequences are obtained from the integers
on the corners of the pyramid
1,
1,
1,
1,
1,
2,
3,
4,
5,
6,
8,
14,
16,
18,
20,
23,
38,
41,
44,
47,
51,
79,
83,
87,
91,
...
...
...
...
...
The first of these is just (4) + 1 and is given by
1
sn = (4n3 + 3n2 − n + 6).
6
The second sequence is new and is given by
1
sn = (4n3 + 3n2 − 25n + 24),
6
while the third is generated by
1
sn = (4n3 + 3n2 − 19n + 18).
6
In general, the second through fifth sequences in (5) are given by
1
sn = (4n3 + 3n2 − 25n + 24) + l(n − 1)
6
for l = 0 to 3. For example, the last sequence is generated by
1
sn = (n + 2)(4n2 − 5n + 3).
6
and (4) is obtained with l = 4.
(5)
824
T. Aaron Gulliver
The sum of the elements on the bottom rows of the pyramid (starting from
the top and moving clockwise), give the sequences
1, 5, 31, 114,
1, 7, 45, 158,
1, 9, 51, 170,
1, 11, 57, 182,
1, 13, 63, 194,
1, 9, 43, 138,
305,
405,
425,
445,
465,
345,
...
...
...
...
...
...
(6)
The first sequence is generated by
1
sn = n(4n3 − 5n2 − 4n + 11)
6
the last is obtained from
1
sn = n(4n3 − 5n2 + 8n − 1)
6
while the remainder are given by
1
sn = n(4n3 + 3n2 − 22n + 21 + 6l(n − 1))
6
for l = 0 to 3.
Now consider rays in the pyramid towards the corners, starting from the
smallest integer on a level. The sum of the elements in the leftmost ray is the
first sequence in (6). The next ray gives the sequence
1, 6, 34, 120, 315, . . .
with terms
1
sn = n(n + 1)(4n2 − 9n + 8).
6
In general, these sequences are generated by
1
sn = n(4n3 − 5n2 + (3l − 4)n − 3l + 11).
6
for l = 0 to 3, so the remaining sequences are
1, 7, 37, 126, 325, . . .
1, 8, 40, 132, 335, . . .
Now consider wedges in the pyramid. The sum of the elements in the
leftmost wedge results in the sequence
1, 9, 72, 320, 1005, . . .
Sequences from hexagonal pyramid of integers
825
with terms
1
n(n + 1)(4n3 − 3n2 − 7n + 12).
12
sn =
The next wedge gives the sequence
1, 11, 80, 340, 1045, . . .
with terms
sn =
1
n(n + 1)(4n3 − 3n2 − 3n + 8).
12
In general, the wedges are given by
sn =
1
n(n + 1)(4n3 − 3n2 + (4l − 7)n − 4l + 12).
12
for l = 0 to 3, so the remaining sequences are
1, 13, 88, 360, 1085, . . .
1, 15, 96, 380, 1125, . . .
Combining the first two wedges gives
1, 14, 118, 540, 1735, . . .
which is generated by
1
sn = n(n + 1)(4n3 − 7n2 + 4n + 2),
6
and adding the next wedge gives
1, 20, 169, 774, 2495, . . .
with
sn =
1
n(12n4 − 13n3 + 2n2 + 13n − 2).
12
Finally, combining the last wedge gives the sum of the elements in each level
1, 27, 225, 1022, 3285, . . .
with
1
sn = n(2n − 1)(4n3 − 3n2 + 2n + 3).
6
826
T. Aaron Gulliver
Now adding the elements on all the levels gives
sn =
n
1
i=1
6
i(2i − 1)(4i3 − 3i2 + 2i + 3)
1
=
n(n + 1)(4n − 1)(4n3 + 3n2 − n + 6)
72
(7)
which gives
1, 28, 253, 1275, 4560, . . .
(8)
This result can also be obtain by summing the positive integers up to the
values in (4)
n(n+1)(4n−1)/6
sn =
i
i=1
(9)
1
n(n + 1)(4n − 1)(4n3 + 3n2 − n + 6).
=
72
In general, the wedge values are given by
sn =
1 n (4l + 4)n4 + (1 − 7l)n3 + (2l2 + 2l − 10)n2
12
+(5 − 3l2 + 10l)n + l2 − 9l + 12)
for l = 0 to 3. Summing these values provides the partial wedge sums
sn
n
1 l (4l + 4)i4 + (1 − 7l)i3 + (2l2 + 2l − 10)i2 + (5 − 3l2 + 10l)i
=
12
i=1
+l2 − 9l + 12)
1
=
n(n + 1) (20 + 20l)n4 + (46 − 2l)n3 + (15l2 − 38l − 56)n2
360
+(98l − 15l2 − 34)n − 78l + 204
(10)
which for l = 0 to 2 is
1, 10, 82, 402, 1407, . . .
1, 15, 133, 673, 2408, . . .
1, 21, 190, 964, 3459, . . .
and (8) for l = 3.
827
Sequences from hexagonal pyramid of integers
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] T.A. Gulliver, Sequences from Pentagonal Pyramids of Integers
[6] N.J.A. Sloane,
On-Line Encyclopedia of Integer
http://www.research.att.com/˜njas/sequences/index.html.
Received: November, 2009
Sequences,