Consider all two digit numbers from 10 to 99 and use these as the seed digits that will generate a third digit by ADDITION of the two digits and by then finding the DIGITAL ROOT of the resultant sum. Here are the 90 starting 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, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999
These in turn will produce 90 four digit numbers. These are:
1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999
These in turn will produce 90 five digit numbers. These are:
10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999
Forgetting about the original two digit numbers, let's group all the three, four and five digits number together so that we have 270 numbers. These are:
101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999
Viewed as a Fibonacci-like sequence, the sequence of digits will eventually cycle. Take 27977 as an example. The progression is:$$2, 7, 9, 7, 7, 5, 3, 8, 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, \dots $$An alternative to this progression of digits is to determine the arithmetical digital root of the cumulative sum of digits and use this as the next digit. Here is a permalink that will generate this sequence of 270 numbers. Here are the numbers:
101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1012, 1124, 1236, 1348, 1451, 1563, 1675, 1787, 1899, 1912, 2024, 2136, 2248, 2351, 2463, 2575, 2687, 2799, 2812, 2924, 3036, 3148, 3251, 3363, 3475, 3587, 3699, 3712, 3824, 3936, 4048, 4151, 4263, 4375, 4487, 4599, 4612, 4724, 4836, 4948, 5051, 5163, 5275, 5387, 5499, 5512, 5624, 5736, 5848, 5951, 6063, 6175, 6287, 6399, 6412, 6524, 6636, 6748, 6851, 6963, 7075, 7187, 7299, 7312, 7424, 7536, 7648, 7751, 7863, 7975, 8087, 8199, 8212, 8324, 8436, 8548, 8651, 8763, 8875, 8987, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10124, 11248, 12363, 13487, 14512, 15636, 16751, 17875, 18999, 19124, 20248, 21363, 22487, 23512, 24636, 25751, 26875, 27999, 28124, 29248, 30363, 31487, 32512, 33636, 34751, 35875, 36999, 37124, 38248, 39363, 40487, 41512, 42636, 43751, 44875, 45999, 46124, 47248, 48363, 49487, 50512, 51636, 52751, 53875, 54999, 55124, 56248, 57363, 58487, 59512, 60636, 61751, 62875, 63999, 64124, 65248, 66363, 67487, 68512, 69636, 70751, 71875, 72999, 73124, 74248, 75363, 76487, 77512, 78636, 79751, 80875, 81999, 82124, 83248, 84363, 85487, 86512, 87636, 88751, 89875, 90999, 91124, 92248, 93363, 94487, 95512, 96636, 97751, 98875, 99999
Let's take 26875 as an example. We begin with 26 as our seed number and then proceed thus: $$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 268 \text{ has digit sum } 16 \equiv 7 &\rightarrow 2687 \\ 2687 \text{ has digit sum } 23 \equiv 5 &\rightarrow 26875 \end{align} $$The three digit numbers are the same as earlier but the differences arise in the four and five digit numbers. Let's compare the previous cumulative results with the seed number 26 again but using the earlier two digit approach:$$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 68 \text{ has digit sum } 14 \equiv 5 &\rightarrow 2685 \\ 85 \text{ has digit sum } 13 \equiv 4 &\rightarrow 26854 \end{align} $$
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