Sequences from Pentagonal Pyramids of Integers

Transcription

1 International Mathematical Fum, 5, 2010, no. 13, Sequences from Pentagonal Pyramids of Integers T. Aaron Gulliver Department of Electrical and Computer Engineering University of Victia, P.O. Box 3055, STN CSC Victia, BC, V8W 3P6, Canada 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 Keywds: 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. F example, the number of elements in the square pyramid is s n = n 2 = n i 2 = 1 n(n + 1)(2n +1), (1) 6 where n is the height of the pyramid. Starting from n = 1, we have 1, 5, 14, 30, 55,..., (2) which is sequence A 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.

2 622 T. Aaron Gulliver 2. Pentagonal Pyramids of Integers A pentagonal pyramidal array of integers has a structure 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 s i =1, 5, 12, 22, 35,... The number of elements in the pyramid is then n 1 2 i(3i 1) = 1 2 n2 (n +1), (3) where n is the height of the pyramid. Starting from n = 1, we have 1, 6, 18, 40, 75,..., (4) which is sequence A [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. F the top two levels, the arrangements are the same 1,

3 Sequences from pentagonal pyramids of integers 623 F the third level, we have and In addition to (4), the following simple sequences are obtained from the integers on the other cners of the first pyramid arrangement 1, 2, 7, 19, 41,... 1, 3, 9, 22, 45,... 1, 4, 12, 28, 55,... 1, 5, 15, 34, 65,... The first of these is sequence A100119, the n-th centered n 1-gonal number, and is given by s n = 1 2 (n3 2n 2 + n +2). The second is sequence A064808, the nth n + 1-gonal number, given by s n = 1 2 n(n2 2n +3), while the third is sequence A047732, the n-th n + 2-gonal number, given by s n = 1 2 n(n2 n +2). The last sequence is A006003, given by s n = n(n2 +1). 2 The cners of the second pyramid arrangement provide the following sequences 1, 2, 7, 19, 41,... 1, 3, 12, 31, 63,... 1, 4, 14, 34, 67,... 1, 5, 16, 37, 71,.....

4 624 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 s n = 1 2 (n3 + n 2 6n + 6) (5) s n = 1 2 (n3 + n 2 4n + 4) (6) s n = 1 2 (n3 + n 2 2n + 2) (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) s n = 1 2 (n3 2n 2 +2n + 1) (9) s n = 1 2 n(n3 n 2 n + 3) (10) respectively. Now consider wedges in the first pyramid. The sum of the elements in the leftmost wedge results in the sequence terms 1, 9, 57, 235, 720,..., s n = 1 8 n(n + 1)(2n3 3n 2 +3n +2). Combining the two leftmost wedges gives sequence A terms 1, 14, 99, 424, 1325,..., s n = n n(n 1)2 + i = 1 2 n2 (n 3 n 2 + n +1).

5 Sequences from pentagonal pyramids of integers 625 This sequence first appeared in [4] as one side of a cube of integers. In this case the expression is s n = n n j=1 n2 (i 1) + j = n 1 2 n(2n2 i 2n 2 + n +1) = 1 2 n2 (n 3 n 2 + n +1) The crespondence between the two shapes is not obvious. Adding the third wedge gives the sum of the numbers on the pyramid level terms 1, 20, 150, 649, 2030,..., s n = 1 8 n(3n 1)(2n3 n 2 + n +2). Now, adding the terms in the above sequence gives the sum of the elements in the entire pyramid This is also equal to 1, 21, 171, 820, 2850,..., i(3i 8 1)(2i3 i 2 + i +2) = 1 8 n2 (n 3 + n 2 +2). s n = 1 2 n2 (n+1) i = 1 8 n2 (n 3 + n 2 +2). The partial sum of the pentagonal pyramidal numbers is sequence A , 7, 25, 65, 140,..., s n = 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 cner elements, i.e. 1, 2+4, , ,..., (11)

6 626 T. Aaron Gulliver which is sequence A [4], Taking the next set of cners gives 1, 6, 30, 100, 255,..., s n = 1 2 n(n + 1)(n2 2n +2). 1, 2+5, , ,... (12) which is sequence A [4], 1, 7, 33, 106, 265,..., s n = 1 2 n(n3 n 2 + n +1). Finally, taking the rightmost cners results in the sequence 1, 2+6, , ,..., (13) which is sequence A [4], 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 2 i(i3 i 2 i +3) = 1n(n + 2 1)(4n3 + n 2 11n + 26). 1, 6, 33, 127, 372,..., i(i + 2 1)(i2 2i +2) = 1 n(n + 1)(n )(12n2 21n + 29).

7 Sequences from pentagonal pyramids of integers 627 which gives the sequence The sum of the elements in (12) is which gives the sequence 1, 7, 37, 137, 392,..., 2 i(i3 i 2 + i +1) = 1 n(n )(12n3 +3n 2 +7n + 38). 1, 8, 41, 147, 412,..., Finally, the sum of the elements in (13) is which gives the sequence 2 i2 (i 2 i +2) = 1 n(n )(4n3 + n 2 +9n +6). 1, 9, 45, 157, 432,..., Returning to the first pyramid of integers, taking the wedges between the rays, we have f the sum of the integers in the first wedge on the left 0, 0, 10, , ,..., (14) The next wedge gives 0, 0, 10, 72, 291,..., s n = 1 8 (n 1)(n 2)(2n3 3n 2 +3n +4). 0, 0, 13, , ,..., (15) 0, 0, 13, 90, 351,..., s n = 1 8 (n + 1)(n 1)(n 2)(2n2 3n +4).

8 628 T. Aaron Gulliver Finally, the last wedge gives 0, 0, 16, , ,..., (16) 0, 0, 16, 108, 411,..., s n = 1(n 1)(n 8 2)(2n3 + n 2 n +4). References [1] T.A. Gulliver, Sequences from Arrays of Integers, Int. Math. J (2002). [2] T.A. Gulliver, Sequences from Integer Tetrahedrons, Int. Math. Fum, 1, (2006). [3] T.A. Gulliver, Sequences from Pyramids of Integers, Int. J. Pure and Applied Math , (2007). [4] T.A. Gulliver, Sequences from Cubes of Integers, Int. Math. J. 4, , (2003). Crection Int. Math. Fum, vol. 1, no, 11, pp [5] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, njas/sequences/index.html. Received: June, 2009