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 agullive@ece.uvic.ca 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 s n =1 2 +2 2 +3 2 +4 2 +5 2 +...+ 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 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 and the resulting integer sequence is i(2i 1) s i =1, 6, 15, 28, 45,... The number of elements in the pyramid is then n i(2i 1) = 1 n(n + 1)(4n 1), (3) 6 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 1, 2 3 7 4 6 5
Sequences from hexagonal pyramid of integers 823 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, 2, 8, 23, 51,... 1, 3, 14, 38, 79,... 1, 4, 16, 41, 83,... 1, 5, 18, 44, 87,... 1, 6, 20, 47, 91,... (5) The first of these is just (4) + 1 and is given by s n = 1 6 (4n3 +3n 2 n +6). The second sequence is new and is given by while the third is generated by s n = 1 6 (4n3 +3n 2 25n + 24), s n = 1 6 (4n3 +3n 2 19n + 18). In general, the second through fifth sequences in (5) are given by s n = 1 6 (4n3 +3n 2 25n + 24) + l(n 1) for l = 0 to 3. For example, the last sequence is generated by and (4) is obtained with l =4. s n = 1 6 (n + 2)(4n2 5n +3).
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, 305,... 1, 7, 45, 158, 405,... 1, 9, 51, 170, 425,... 1, 11, 57, 182, 445,... 1, 13, 63, 194, 465,... 1, 9, 43, 138, 345,... (6) The first sequence is generated by the last is obtained from while the remainder are given by s n = 1 6 n(4n3 5n 2 4n + 11) s n = 1 6 n(4n3 5n 2 +8n 1) s n = 1 6 n(4n3 +3n 2 22n +21+6l(n 1)) 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 with terms 1, 6, 34, 120, 315,... s n = 1 6 n(n + 1)(4n2 9n +8). In general, these sequences are generated by s n = 1 6 n(4n3 5n 2 +(3l 4)n 3l + 11). 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. leftmost wedge results in the sequence The sum of the elements in the 1, 9, 72, 320, 1005,...
Sequences from hexagonal pyramid of integers 825 with terms s n = 1 12 n(n + 1)(4n3 3n 2 7n + 12). The next wedge gives the sequence with terms In general, the wedges are given by 1, 11, 80, 340, 1045,... s n = 1 12 n(n + 1)(4n3 3n 2 3n +8). s n = 1 12 n(n + 1)(4n3 3n 2 +(4l 7)n 4l + 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 which is generated by and adding the next wedge gives with 1, 14, 118, 540, 1735,... s n = 1 6 n(n + 1)(4n3 7n 2 +4n +2), 1, 20, 169, 774, 2495,... s n = 1 12 n(12n4 13n 3 +2n 2 +13n 2). Finally, combining the last wedge gives the sum of the elements in each level with 1, 27, 225, 1022, 3285,... s n = 1 6 n(2n 1)(4n3 3n 2 +2n +3).
826 T. Aaron Gulliver Now adding the elements on all the levels gives n 1 s n = 6 i(2i 1)(4i3 3i 2 +2i +3) = 1 72 n(n + 1)(4n 1)(4n3 +3n 2 n +6) (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) s n = n(n+1)(4n 1)/6 i = 1 72 n(n + 1)(4n 1)(4n3 +3n 2 n +6). (9) In general, the wedge values are given by s n = 1 12 n [ (4l +4)n 4 +(1 7l)n 3 +(2l 2 +2l 10)n 2 +(5 3l 2 +10l)n + l 2 9l + 12) ] for l = 0 to 3. Summing these values provides the partial wedge sums n 1 s n = 12 l [ (4l +4)i 4 +(1 7l)i 3 +(2l 2 +2l 10)i 2 +(5 3l 2 +10l)i +l 2 9l + 12) ] = 1 360 n(n +1)[ (20 + 20l)n 4 + (46 2l)n 3 + (15l 2 38l 56)n 2 +(98l 15l 2 34)n 78l + 204 ] (10) which for l = 0 to 2 is and (8) for l =3. 1, 10, 82, 402, 1407,... 1, 15, 133, 673, 2408,... 1, 21, 190, 964, 3459,...
Sequences from hexagonal pyramid of integers 827 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 Sequences, http://www.research.att.com/ njas/sequences/index.html. Received: November, 2009