Subtractive Fibonacci-like Numbers

Consider all two digit numbers from 10 to 99 and use these as the seed digits that will generate a third number NOT by ADDITION of the two digits but by SUBTRACTION, subtracting the smaller digit from the larger when they are different. This ensures that the result is always positive or zero. Taking the absolute value of the result is another way to regard it. Here are the 90 numbers.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

These 90 two digit numbers will generate another 90 three digit numbers. These are:

101, 110, 121, 132, 143, 154, 165, 176, 187, 198, 202, 211, 220, 231, 242, 253, 264, 275, 286, 297, 303, 312, 321, 330, 341, 352, 363, 374, 385, 396, 404, 413, 422, 431, 440, 451, 462, 473, 484, 495, 505, 514, 523, 532, 541, 550, 561, 572, 583, 594, 606, 615, 624, 633, 642, 651, 660, 671, 682, 693, 707, 716, 725, 734, 743, 752, 761, 770, 781, 792, 808, 817, 826, 835, 844, 853, 862, 871, 880, 891, 909, 918, 927, 936, 945, 954, 963, 972, 981, 990

These in turn will spawn another 90 numbers. These are:

1011, 1101, 1211, 1321, 1431, 1541, 1651, 1761, 1871, 1981, 2022, 2110, 2202, 2312, 2422, 2532, 2642, 2752, 2862, 2972, 3033, 3121, 3211, 3303, 3413, 3523, 3633, 3743, 3853, 3963, 4044, 4132, 4220, 4312, 4404, 4514, 4624, 4734, 4844, 4954, 5055, 5143, 5231, 5321, 5413, 5505, 5615, 5725, 5835, 5945, 6066, 6154, 6242, 6330, 6422, 6514, 6606, 6716, 6826, 6936, 7077, 7165, 7253, 7341, 7431, 7523, 7615, 7707, 7817, 7927, 8088, 8176, 8264, 8352, 8440, 8532, 8624, 8716, 8808, 8918, 9099, 9187, 9275, 9363, 9451, 9541, 9633, 9725, 9817, 9909

These in turn will spawn another 90 numbers. These are:

10110, 11011, 12110, 13211, 14312, 15413, 16514, 17615, 18716, 19817, 20220, 21101, 22022, 23121, 24220, 25321, 26422, 27523, 28624, 29725, 30330, 31211, 32110, 33033, 34132, 35231, 36330, 37431, 38532, 39633, 40440, 41321, 42202, 43121, 44044, 45143, 46242, 47341, 48440, 49541, 50550, 51431, 52312, 53211, 54132, 55055, 56154, 57253, 58352, 59451, 60660, 61541, 62422, 63303, 64220, 65143, 66066, 67165, 68264, 69363, 70770, 71651, 72532, 73413, 74312, 75231, 76154, 77077, 78176, 79275, 80880, 81761, 82642, 83523, 84404, 85321, 86242, 87165, 88088, 89187, 90990, 91871, 92752, 93633, 94514, 95413, 96330, 97253, 98176, 99099

Let's forget about our two digit starting numbers and consider only the resulting three, four and five digit numbers. Grouping them all together we have the following 270 member sequence:

101, 110, 121, 132, 143, 154, 165, 176, 187, 198, 202, 211, 220, 231, 242, 253, 264, 275, 286, 297, 303, 312, 321, 330, 341, 352, 363, 374, 385, 396, 404, 413, 422, 431, 440, 451, 462, 473, 484, 495, 505, 514, 523, 532, 541, 550, 561, 572, 583, 594, 606, 615, 624, 633, 642, 651, 660, 671, 682, 693, 707, 716, 725, 734, 743, 752, 761, 770, 781, 792, 808, 817, 826, 835, 844, 853, 862, 871, 880, 891, 909, 918, 927, 936, 945, 954, 963, 972, 981, 990, 1011, 1101, 1211, 1321, 1431, 1541, 1651, 1761, 1871, 1981, 2022, 2110, 2202, 2312, 2422, 2532, 2642, 2752, 2862, 2972, 3033, 3121, 3211, 3303, 3413, 3523, 3633, 3743, 3853, 3963, 4044, 4132, 4220, 4312, 4404, 4514, 4624, 4734, 4844, 4954, 5055, 5143, 5231, 5321, 5413, 5505, 5615, 5725, 5835, 5945, 6066, 6154, 6242, 6330, 6422, 6514, 6606, 6716, 6826, 6936, 7077, 7165, 7253, 7341, 7431, 7523, 7615, 7707, 7817, 7927, 8088, 8176, 8264, 8352, 8440, 8532, 8624, 8716, 8808, 8918, 9099, 9187, 9275, 9363, 9451, 9541, 9633, 9725, 9817, 9909, 10110, 11011, 12110, 13211, 14312, 15413, 16514, 17615, 18716, 19817, 20220, 21101, 22022, 23121, 24220, 25321, 26422, 27523, 28624, 29725, 30330, 31211, 32110, 33033, 34132, 35231, 36330, 37431, 38532, 39633, 40440, 41321, 42202, 43121, 44044, 45143, 46242, 47341, 48440, 49541, 50550, 51431, 52312, 53211, 54132, 55055, 56154, 57253, 58352, 59451, 60660, 61541, 62422, 63303, 64220, 65143, 66066, 67165, 68264, 69363, 70770, 71651, 72532, 73413, 74312, 75231, 76154, 77077, 78176, 79275, 80880, 81761, 82642, 83523, 84404, 85321, 86242, 87165, 88088, 89187, 90990, 91871, 92752, 93633, 94514, 95413, 96330, 97253, 98176, 99099

The sequence will continue indefinitely and what drew my attention to these types of numbers was the number associated with my diurnal age today, 27520. I noticed that it almost qualified because 7 - 2 = 5, 7 - 5 = 2 but 5 - 2 does not produce the required final digit of 3. However, in three more days it will when my diurnal age reaches 27523 days. The next number in the sequence is 28624 which is some three years away. Such numbers are certainly not frequent so they deserve to be given some attention.

Viewed as a Fibonacci-like sequence of numbers, the sequences all settle down to a repetitive \(0, n, n\) pattern where \(n\) is a digit between 1 and 9. For example, 27523 becomes:$$2, 7, 5, 2, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, \dots$$The fact that the absolute value of the repeated subtraction of the two digits cannot produce numbers greater than 9 ensures eventual repetition. Subtraction aside, there are similar sequences to explore like adding the first two digits together to produce a third number that is not the sum of the first two, as in the Fibonacci sequence, but the DIGITAL ROOT of the number. For example:$$ \begin{align} 27 &\rightarrow 279 \\279 &\rightarrow 2797 \text{ since } 9 + 7 = 16 \rightarrow 7 \\ 2797 &\rightarrow 27977 \end{align} $$This sequence will be examined in my next post.