Consolidating Fibonacci-like Numbers

In my post titled Additive Fibonacci-like Numbers I was dealing with additive digital roots to generate additional digits after the starting two digits were in place. For example, let's start with 78:$$ \begin{align} 78 \rightarrow 7 + 8 =15 \rightarrow 1+5=6 &\rightarrow 786 \\786 \rightarrow 8+6=14 \rightarrow 1+4=5 &\rightarrow 7865 \\7865 \rightarrow 6+5 =11 \rightarrow 1+1=2 &\rightarrow 78652 \end{align} $$We could keep going forever. The advantage of this approach is that the sum of the two previous digits reduces to a single digit between 1 and 9. Let's call these types of numbers Additive Fibonacci-like Numbers of the First Type. Between 100 and 1,000,000 these numbers 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

However, in my previous post, Variations on the Taxi Cab Number, I was not working with the digital roots and this is a severe limitation. The early digits need to be small if the digits are to progress in a Fibonacci-like manner. That's why, in the range of numbers, up to one million, the largest number is 303369. This number is constructed as follows beginning with the first two digits 3 and 0:$$ \begin{align} 30 \rightarrow 3 + 0 &= 3 \rightarrow 303 \\ 303 \rightarrow 0+3 &=3 \rightarrow 3033\\3033 \rightarrow 3 + 3 &= 6 \rightarrow 30336\\30336 \rightarrow 3+6 &= 9 \rightarrow 303369 \end{align}$$We can't go any further because of the final two digits: 6 + 9 = 15. Let's call these types of numbers Additive Fibonacci-like Numbers of the Second Type. Between 100 and 1,000,000 these numbers are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1011, 1123, 1235, 1347, 1459, 2022, 2134, 2246, 2358, 3033, 3145, 3257, 3369, 4044, 4156, 4268, 5055, 5167, 5279, 6066, 6178, 7077, 7189, 8088, 9099, 10112, 11235, 12358, 20224, 21347, 30336, 31459, 40448, 101123, 112358, 202246, 303369

With bases higher than 10, the 1 to 9 digit limitation can be exceeded. For example in base 16, if we start as before with an initial 78 then a third digit is possible:$$78 \rightarrow 7 + 8 = 15 = F \rightarrow 78F$$Thus we have:$$ \begin{align} 78F_{16} &= 7 \times 16^2 + 8 \times 16 + 15 \\ &=1935_{10} \end{align} $$This means that 1935 is an Additive Fibonacci-like Number of the Second Type in base 16. Here is a list of numbers greater than 27000 and less than 40000 that are "additive Fibonacci-like" and of the "second type" in base 16 (permalink):

• 28791 --> 7077
• 29065 --> 7189
• 29339 --> 729b
• 29613 --> 73ad
• 29887 --> 74bf
• 32904 --> 8088
• 33178 --> 819a
• 33452 --> 82ac
• 33726 --> 83be
• 37017 --> 9099
• 37291 --> 91ab
• 37565 --> 92bd
• 37839 --> 93cf

Additive Fibonacci-like Numbers of the Second Type in base 16 are thus:

28791, 29065, 29339, 29613, 29887, 32904, 33178, 33452, 33726, 37017, 37291, 37565, 37839

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Here the numbers greater than 27000 and less than 40000 for base 15:

• 27128 --> 8088
• 27370 --> 819a
• 27612 --> 82ac
• 27854 --> 83be
• 30519 --> 9099
• 30761 --> 91ab
• 31003 --> 92bd
• 33910 --> a0aa
• 34152 --> a1bc
• 34394 --> a2ce
• 37301 --> b0bb
• 37543 --> b1cd

Additive Fibonacci-like Numbers of the Second Type in base 15 are (permalink):

27128, 27370, 27612, 27854, 30519, 30761, 31003, 33910, 34152, 34394, 37301, 37543

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Here are the numbers greater than 27000 and less than 40000 for base 14 (permalink): 

• 27590 --> a0aa
• 27802 --> a1bc
• 30349 --> b0bb
• 30561 --> b1cd
• 33108 --> c0cc
• 35867 --> d0dd
• 38628 --> 10112

Additive Fibonacci-like Numbers of the Second Type in base 14 are thus:

27590, 27802, 30349, 30561, 33108, 35867, 38628

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Here are the numbers greater than 27000 and less than 40000 for base 13 (permalink):

• 28745 --> 10112
• 31140 --> 11235
• 33535 --> 12358
• 35930 --> 1347b

Additive Fibonacci-like Numbers of the Second Type in base 13 are thus:

28745, 31140, 33535, 35930

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For base 12, there are none between 27000 and 40000 but for base 11 we have (permalink):

• 29550 --> 20224
• 31027 --> 21347
• 32504 --> 2246a

Additive Fibonacci-like Numbers of the Second Type in base 11 are thus:

29550, 31027, 32504

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Just for completeness I'll now look at bases 10 and lower. For base 10, we have (permalink):

• 30336 --> 30336
• 31459 --> 31459

Additive Fibonacci-like Numbers of the Second Type in base 10 are thus:

30336, 31459

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For base 9 there are none but for base 8 there is one (permalink):

• 33363 --> 101123

Additive Fibonacci-like Numbers of the Second Type in base 8 are thus :

33363

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For base 7, we have 

• 34432 --> 202246

Additive Fibonacci-like Numbers of the Second Type in base 7 are thus:

34432

There are no suitable numbers in the range 27000 to 40000 for bases 2, 3, 4, 5 and 6. I've added this determination of whether a number is additive Fibonacci-like of the second type to my multipurpose algorithm.

RIGHT TO LEFT INSTEAD OF LEFT TO RIGHT

There's no compulsion to proceed from left to right when working with digits and so a new set of numbers can be generated by simply reversing the order of the digits. Thus Additive Fibonacci-like Numbers of the First Type are shown below where digit progression is from right to left:

101, 119, 128, 137, 146, 155, 164, 173, 182, 191, 202, 211, 229, 238, 247, 256, 265, 274, 283, 292, 303, 312, 321, 339, 348, 357, 366, 375, 384, 393, 404, 413, 422, 431, 449, 458, 467, 476, 485, 494, 505, 514, 523, 532, 541, 559, 568, 577, 586, 595, 606, 615, 624, 633, 642, 651, 669, 678, 687, 696, 707, 716, 725, 734, 743, 752, 761, 779, 788, 797, 808, 817, 826, 835, 844, 853, 862, 871, 889, 898, 909, 918, 927, 936, 945, 954, 963, 972, 981, 999, 1101, 1191, 1283, 1375, 1467, 1559, 1642, 1734, 1826, 1918, 2119, 2202, 2292, 2384, 2476, 2568, 2651, 2743, 2835, 2927, 3128, 3211, 3303, 3393, 3485, 3577, 3669, 3752, 3844, 3936, 4137, 4229, 4312, 4404, 4494, 4586, 4678, 4761, 4853, 4945, 5146, 5238, 5321, 5413, 5505, 5595, 5687, 5779, 5862, 5954, 6155, 6247, 6339, 6422, 6514, 6606, 6696, 6788, 6871, 6963, 7164, 7256, 7348, 7431, 7523, 7615, 7707, 7797, 7889, 7972, 8173, 8265, 8357, 8449, 8532, 8624, 8716, 8808, 8898, 8981, 9182, 9274, 9366, 9458, 9541, 9633, 9725, 9817, 9909, 9999, 11918, 12835, 13752, 14678, 15505, 15595, 16422, 17348, 18265, 19182, 21101, 21191, 22927, 23844, 24761, 25687, 26514, 27431, 28357, 29274, 31283, 32119, 33936, 34853, 35779, 36606, 36696, 37523, 38449, 39366, 41375, 42202, 42292, 43128, 44945, 45862, 46788, 47615, 48532, 49458, 51467, 52384, 53211, 54137, 55954, 56871, 57707, 57797, 58624, 59541, 61559, 62476, 63303, 63393, 64229, 65146, 66963, 67889, 68716, 69633, 71642, 72568, 73485, 74312, 75238, 76155, 77972, 78808, 78898, 79725, 81734, 82651, 83577, 84404, 84494, 85321, 86247, 87164, 88981, 89817, 91826, 92743, 93669, 94586, 95413, 96339, 97256, 98173, 99909, 99999

Similarly Additive Fibonacci-like Numbers of the Second Type are shown below where digit progression is from right to left:

101, 202, 211, 303, 312, 321, 404, 413, 422, 431, 505, 514, 523, 532, 541, 606, 615, 624, 633, 642, 651, 707, 716, 725, 734, 743, 752, 761, 808, 817, 826, 835, 844, 853, 862, 871, 909, 918, 927, 936, 945, 954, 963, 972, 981, 1101, 2202, 3211, 3303, 4312, 4404, 5321, 5413, 5505, 6422, 6514, 6606, 7431, 7523, 7615, 7707, 8532, 8624, 8716, 8808, 9541, 9633, 9725, 9817, 9909, 21101, 42202, 53211, 63303, 74312, 84404, 85321, 95413, 321101, 642202, 853211, 963303

If we proceed from left to right, then the third digit is the difference between the first and second digits and so on (this is the subtraction sequence mentioned in my previous post).