Table as a function of x and y

1. Dec 13, 2007

ramsey2879

Each cell value of the x-y table below is (x^2 -2xy + y^2 -x -y)/2. The columns, rows and sloping diagonals have the recursive formula a(n) = 2 *a(n-1) – a(n-2) + d^2, d = the delta y plus delta x. A property is that any pair of adjacent cell values of a row or column when used as arguments in the original formula used to compute their values returns the row or column number from which the values are found. Thus the table itself predicts the cell values of other cells without doing the calculations. Thus since the adjacent pair 01 04 appears in row 2 , the value 2 appears in column 1, row 4 and in row 1, column 4. Since the pairs 03 06 and 02 06 are in rows 0 and 4 respectively, row 6 would have a 0 in column 3 and a 4 in column 2 if it were shown.
There is also a simple method to change the cell values in each row to form pair couples which when used in calculation return a multiple of the row number. If instead of obtaining the row number by the calculation you want the calculation to give m times the row number you would add (m-1)(m-2y)/2 to each cell value.

Thus to convert the values in columns 3 and 2 to ordered pairs that give the row number times 7, you would add 6*(7-6)/2 or 3 to the values of column 3 and 6*(7-4)/2 or 9 to the values in column 2
(12, 8) -> -2 but (15,17) -> -14
(7,4) -> -1 but (10,13) -> -7
(3,1) -> 0 and (6,10) -> 0
(0,-1) -> 1 but (3,8) -> 7
(-2,-2) -> 2 but (1,7) -> 14
(-3,-2) -> 3 but (0,7) -> 21
(-3,-1) -> 4 but (0,8) -> 28
(-2,1) -> 5 but (1,10) -> 35

x\y 03 02 01 00 -1 -2 -3 -4
-------------------------------
-2.. 12 08 05 03 02 02 03 05
-1.. 07 04 02 01 01 02 04 07
00.. 03 01 00 00 01 03 06 10
01.. 00 -1 -1 00 02 05 09 14
02.. -2 -2 -1 01 04 08 13 19
03.. -3 -2 00 03 07 12 18 25
04.. -3 -1 02 06 11 17 24 32
05.. -2 01 05 10 16 23 31 40

Last edited: Dec 13, 2007