Sequences from Centered Hexagons of Integers

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1 International Mathematical Forum, 4, 009, no. 39, Sequences from Centered Hexagons of Integers T. Aaron Gulliver Department of Electrical and Computer Engineering University of Victoria, P.O. Box 3055, STN CSC Victoria, BC, Canada V8W 3P6 Abstract This paper presents a number of sequences based on integers arranged in a centered hexagon 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 In a previous paper [1], many new integer sequences were obtained from two-dimensional arrays of integers. In this paper, we consider arrays of integers in a hexagonal (six-sided) shape. A figurate number is a number that can be represented as a regular geometric pattern. The centered hexagonal numbers are a class of figurate numbers. The figures are formed by a central element, surrounded by concentric hexagons. Each side of a ring contains one more element than a side in the previous ring. Here the elements are consecutive integers, starting with 1 in the centre and increasing along each ring. Thus the first ring has, 3, 4, 5, 6, 7, and these numbers will be used to denote the vertices of the array. An array with n rings contains s n = 6n +6n + elements. The first numbers in the corresponding sequence (starting from n = 0), are 1, 7, 19, 37, 61, 91,...

2 1950 T. Aaron Gulliver which is sequence A00315 in the Encyclopedia of Integer Sequences maintained by Sloane []. The generating function of these centered hexagonal numbers is 1+4x + x =1+7x +19x +37x (1 x) 3 This sequence represents the maximum integer in the n-th ring of the array. Note that taking the sum of the first n centered hexagonal numbers gives n 1 i=0 s i = n 1 i=0 6i +6i + = n 3. Hexagonal Arrays of Numbers Starting at the k-th vertex, one can form six sequences of integers given by n s n =+(k 8)n + =3n +(k 5)n + (1) i=1 For k = to 7 the sequences are, 8, 0, 38, 6,... 3, 10, 3, 4, 67,... 4, 1, 6, 46, 7,... 5, 14, 9, 50, 77,... 6, 16, 3, 54, 8,... 7, 18, 35, 58, 87,... The first sequence is A077588, the maximum number of regions the plane is divided into by n triangles. The third sequence is A07599, while the fourth is A005918, the number of points on the surface of a square pyramid. The other sequences are new. Taking the sum of the integers starting from each vertex gives 3n +(k 5)n += l(l +(k )l + k) () For k = to 7 the sequences are, 10, 30, 68, 130,... 3, 13, 36, 78, 145,... 4, 16, 4, 88, 160,... 5, 19, 48, 98, 175,... 6,, 54, 108, 190,... 7, 5, 60, 118, 05,... The first sequence is A0346 in [], while the others are new.

3 Sequences from centered hexagons of integers 1951 Summing the elements in the n-th ring of the array gives s n = 6n(6n +3) (3) which has terms 7, 16, 513, 1188, 95,... The sum of the elements in all the rings (including ring 0) is then 1+ 6n(6n +3) which has terms =1+ 9l(l + 1)(l + l +1) 1, 8, 190, 703, 1891,... = (3l +3l + )(3l +3l +1) (4) This is sequence A in []. Note that the sequence is given as 9n 4 +18n +5 as n runs through the odd integers, but letting n =l + 1 provides (4). The remainder of the sequences in this paper are new. One can also takes wedges in the arrays. For example, the sequence for the wedge between (but including) vertices and 3 is + 3 = 5, = 7, = 86,... The sequences in increasing wedge size of to k are generated by s n = ((k )n + 1)(6n +(k 8)n +4) (5) For k = 3 to 7 these sequences are 5, 7, 86, 00, 387,... 9, 50, 161, 378, 737,... 14, 77, 45, 57, 111,... 0, 108, 338, 78, 151,... 7, 143, 440, 1008, 1937,... Since the last sequence represents just five wedges of the array, there is a difference between it and (3). For the first ring, they are the same, while the difference for the second ring is 19 (just the integer between vertices and 7 in the ring). The sequence for the differences is (n 1)(6n +5n +4) (6)

4 195 T. Aaron Gulliver 0, 19, 73, 180, 358,... Taking the sums of the wedge elements gives ((k )n + 1)(6n +(k 8)n +4) (7) = l 1 (5k + k 6kl +8l +k l +3k l kl +9kl 3 18l 3 ) For k = 3 to 7 the sequences are 5, 3, 118, 318, 705,... 9, 59, 0, 598, 1335,... 14, 91, 336, 908, 00,... 0, 18, 466, 148, 760,... 7, 170, 610, 1618, 3555,... The sequence for the sum of the differences (6) is (n 1)(6n +5n +4) = l(l 1)(9n +5l + 8) (8) 0, 19, 9, 7, 630,... One can also form sequences from other wedges in the arrays. Between vertices 3 and 4 we have Between vertices 4 and 5 we have s n = (n + 1)(6n 3n +4) 7, 33, 98, 0, 417,... s n = (n + 1)(6n n +4) 9, 39, 110, 40, 447,... (9) (10) Generalizing (9) and (10), the sequences for the wedges between adjacent vertices k and k + 1 are s n = (n + 1)(6n +kn 9n +4) (11)

5 Sequences from centered hexagons of integers 1953 For k = 5, this gives the sequence and for k = 6 the sequence 11, 45, 1, 60, 477,... 13, 51, 134, 80, 507,... Taking the sums of the elements in the wedges between adjacent vertices gives (n + 1)(6n +kn 9n +4) (1) = l 1 (9l3 +1l +4kl 15l +1kl +8k +6) For k = 3 to 6 the sequences are 7, 40, 138, 358, 775,... 9, 48, 158, 398, 845,... 11, 56, 178, 438, 915,... 13, 64, 198, 478, 985,... The sequence for k = was given previously. For pairs of wedges, the sequences starting at vertex k are given by s n = (n + 1)(6n +(k 8)n +4) For k =, the sequence was given previously. For k = 3 to 5, we have (13) 1, 60, 18, 414, 79,... 15, 70, 03, 450, 847,... 18, 80, 4, 486, 90,... Taking the sums of the elements in the wedge pairs between adjacent vertices gives (n + 1)(6n +(k 8)n +4) (14) For k = 3 to 5 the sequences are = l 6 (9l3 +8l +4kl +9kl 6l +5k +7) 1, 7, 54, 668, 1460,... 15, 85, 88, 738, 1585,... 18, 98, 3, 808, 1710,...

6 1954 T. Aaron Gulliver The sequence for k = was given previously. Sequence (6) gives the values between vertices and 7. between other pairs of vertices are generated by s n = (n 1)(6n +(k 9)n +4) The sequences (15) For k = 7, we obtain (6). For k =, 3, 4, 5 and 6, we have 0, 9, 43, 10, 58,... 0, 11, 49, 13, 78,... 0, 13, 55, 144, 98,... 0, 15, 61, 156, 318,... 0, 17, 67, 168, 338,... Taking the sums of the elements between vertices gives (n 1)(6n +(k 9)n +4) (16) For k = to 6 the sequences are = l 1 (l 1)(9l +4kl 3l +4k) 0, 9, 5, 17, 430,... 0, 11, 60, 19, 470,... 0, 13, 68, 1, 510,... 0, 15, 76, 3, 550,... 0, 17, 84, 5, 590,... The sequence for k = 7 was given previously. References [1] T.A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal (00). [] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, njas/sequences/index.html. Received: October, 008