The following array is based on the Fibonacci series. The first row is simply the Fibonacci series. Each nth row are Fibonacci type series where the generating numbers in columns 1 and 2 are determined as follows. The numbers in column 1 are the row number and the number to the right of the row number in column 2 is equal to 1 plus the number to the right of the row number where it appears in columns 3 and above. For instance row numbers 1 –3 are found in columns 3-5 at the top of the table. Therefore the numbers to the right of the row number in column 2 are equal to the next higher Fibonacci number plus 1. This table will generate each of the natural numbers only once in columns 3 and above so the array is precisely defined. If anyone could prove this fact I would appreciate it. The number to the right of 7 in row 1 is 11. Therefore 12 is placed in column 2 of row 7. There is a magical property to this table give the simple rule for its formation that goes far beyond the fact that every natural number appears only once in columns 3 and above. Note that rows 2-4 are the same as Row 0 (or the Fibonacci series) multiplied respectively by 2 through 4 offset by 2. Rows 3*5, 3*6 … 3*11 are the same as the Fibonacci series multiplied by 5-11. This can be seen by looking at the numbers to the right of 15, 18, ... 33 where they appear in columns 3 and above and verifying that they are each 1 less than 8*5,8*6,8*7...8*11. Also, rows 4*2 and 4*3 are the same as row 1 multiplied by 2 and 3 respectively. There is an obvious pattern here which is magical given the simple rule for forming the array. Since 1 and 3 are the 3rd and 5 term of the "0" row and 4 is the 3rd term of the 1st row but row 4*4 is not a multiple of the first row, I suspected that since 11 is the third term of row 1, then rows 4*11 and 5*11 could be the same as row 1 multiplied by 4 and 5 respectively. Yes!! Note that 71 appears to the right of 44 in row 10 and that the next term after 55 in the Fibonacci series is 89. Now 71+1 = 4*18 and 89+1 = 5*18 and 18 appears to the right of 11 in row 1. Thus rows 4*11 and 5*11 are indeed as predicted!!. After more investigating, rows 21*12 through 21* 29 were found to be the same as row 0 multiplied by 12-29 respectively but that 55*30 was the multiple of row 0 by 30. 4, 11, and 29 are odd terms in row 1!! The amazing properties continue with multiples of rows 5 and 6, in fact of any row (except lacking the first few terms) appearing also. Has anyone seen anything like this posted before? 0 1 1 2 3 5 .... 1 3 4 7 11 18 .... 2 4 6 10 16 26 ... 3 6 9 15 24 39 ... 4 8 12 20 32 52 ... 5 9 14 23 37 60 ... 6 11 17 28 45 73 ... 7 12 19 31 50 81 ... 8 14 22 36 58 94 ... 9 16 25 41 66 107 ... 10 17 27 44 71 115 ... 11 19 30 49 79 128 ... 12 21 33 54 87 141 ... 13 22 35 57 92 149 ... 14 24 38 62 100 162 ... 15 25 40 65 105 170 ... 16 27 43 70 113 183 ... 17 29 46 75 121 196 ... 18 30 48 78 126 204 ... ... ...
Please don't use the enter key when posting unless you're writing code, equations, or using paragraphs (which you should, by the way). As for the pattern, I'm sure it wouldn't be too difficult to prove it was true.