The Function x^2 + xy + y^2 – Part 4

4)      Rows, columns, and diagonals in the table of x² + xy + y² follow patterns.

Each row, column, and diagonal (bottom left to top right if x and y are both positive) of x² + xy + y² follow the same pattern, where the difference between two numbers is two more than the difference between the next two numbers.  For example, in the first row (3, 7, 13, 21, 31, 43, 57, 73, 91, 111), 7 – 3 = 4, 13 – 7 = 6, 21 – 13 = 8, 31 – 21 = 10, 43 – 31 = 12, 57 – 43 = 14, 73 – 57 = 16, 91 – 73 = 18, and 111 – 91 = 20.  These differences (4, 6, 8, 10, 12, 14, 16, 18, and 20) are all 2 apart.  In the fourth column (21, 28, 37, 48, 61, 76, 93, 112, 133, and 156), 28 – 21 = 7, 37 – 28 = 9, 48 – 37 = 11, 61 – 48 = 13, 76 – 61 = 15, 93 – 76 = 17, 112 – 93 = 19, and 133 – 112 = 21.  These differences (7, 9, 11, 13, 15, 17, and 19) are also all 2 apart.  In the diagonal (111, 103, 97, 93, 91, 91, 93, 97, 103, and 111), 103 – 111 = -8, 97 – 103 = -6, 93 – 97 = -4,  91 – 93 = -2, 91 – 91 = 0, 93 – 91 = 2, 97 – 93 = 4, 103 – 97 = 6, 111 – 103 = 8.  These differences (-8, -6, -4, -2, 0, 2, 4, 6, and 8) are also all 2 apart.

	1	2	3	4	5	6	7	8	9	10
1	3	7	13	21	31	43	57	73	91	111
2	7	12	19	28	39	52	67	84	103	124
3	13	19	27	37	49	63	79	97	117	139
4	21	28	37	48	61	76	93	112	133	156
5	31	39	49	61	75	91	109	129	151	175
6	43	52	63	76	91	108	127	148	171	196
7	57	67	79	93	109	127	147	169	193	219
8	73	84	97	112	129	148	169	192	217	244
9	91	103	117	133	151	171	193	217	243	271
10	111	124	139	156	175	196	219	244	271	300

To prove this for each row of numbers, the difference between two consecutive numbers (x + 1, y) – (x, y) = ((x + 1)² + (x + 1)y + y²) – (x² + xy + y²) = (x² + 2x + 1 + xy + y + y²) – (x² + xy + y²) = 2x + y + 1.  The difference of the next two consecutive numbers (x + 2, y) – (x + 1, y) = ((x + 2)² + (x + 2)y + y²) – ((x + 1)² + (x + 1)y + y²) = (x² + 4x + 4 + xy + 2y + y²) – (x² + 2x + 1 + xy + y + y²) = 2x + y + 3.  The difference of those differences is (2x + y + 3) – (2x + y + 1) = 2.  A similar argument can be made for each column of numbers, but switching x and y.  For each diagonal of numbers, the difference between two consecutive numbers is (x + 1, y – 1) – (x, y) = ((x + 1)² + (x + 1)(y – 1) + (y – 1)²) – (x² + xy + y²) = (x² + 2x + 1 + xy – x + y – 1 + y² – 2y + 1) – (x² + xy + y²) = 2x + 1 – x + y – 1 – 2y + 1 = x – y + 1.  The difference between the next two consecutive numbers (x + 2, y – 2) – (x + 1, y – 1) = ((x + 2)² + (x + 2)(y – 2) + (y – 2)²) – ((x + 1)² + (x + 1)(y – 1) + (y – 1)²) = (x² + 4x + 4 + xy – 2x + 2y – 4 + y² – 4y + 4) – (x² + 2x + 1 + xy – x + y – 1 + y² – 2y + 1) = x – y + 3.  The differences of those differences is (x – y + 3) – (x – y + 1) = 2.