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:$$ 78 \rightarrow 7 + 8 =15 \rightarrow 1+5=6$$This gives 786 and now:$$86 \rightarrow 8+6=14 \rightarrow 1+4=5$$Thus the new number is 7865 and so in summary:$$78 \rightarrow 786 \rightarrow 7865$$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. 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, this limitation is reduced. For example in base 16, if we start as before with an initial 78 then a third digit is possible:$$78 \rightarrow 7 + 8 = 15 = F$$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:

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:

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:

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:

29550, 31027, 32504

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For base 10, we have (permalink):

• 30336 --> 30336
• 31459 --> 31459

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

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:

33363

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For base 7, we have 

• 34432 --> 202246

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

34432

There are none 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.

A Variation on the Taxi Cab Number

The number 1729 is famous as the so-called "taxi cab number" in memory of the interchange between the mathematicians Hardy and Ramanujan in which the latter observed that the number of the taxi cab in which the former had arrived at the hospital was far from boring (as Hardy had thought). Instead 1729 is the first positive integer that is the sum of two positive cubes in two different ways:$$ \begin{align} 1729 &= 1^3+12^3\\ &= 9^3+10^3 \end{align} $$Today I observed a taxi with the number plate T 3257 and noted that there is a touch of Fibonacci about its digits because:$$ 3 +2 = 5 \text{ and } 2 + 5 = 7$$The digits thus form a Fibonacci-type sequence:$$ 3, 2, 5, 7$$This got me thinking about what numbers with three or more digit have this Fibonacci-like property. Well, up to one million, there are only 82 such numbers so they form a rather exclusive club. 

Here they are (permalink) in a sequence that we'll called the ADDITION SEQUENCE:

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

I have made a related post titled Additive Fibonacci-like Numbers on the 7th August 2024 but this involved finding the digital roots of numbers unlike what I've done here. So for me 3257 will remain my personal taxi cab number.

Another sequence will emerge if, instead of adding the second number to the first and so on, we SUBTRACT the second from the first and so on. In this scenario, 3211 would satisfy because:$$  3 - 2 = 1 \text{ and } 2 -1 =1$$Up to one million, there are 99 such numbers of three digits or more. Here they are in a sequence we'll call the SUBTRACTION SEQUENCE (permalink):

101, 110, 202, 211, 220, 303, 312, 321, 330, 404, 413, 422, 431, 440, 505, 514, 523, 532, 541, 550, 606, 615, 624, 633, 642, 651, 660, 707, 716, 725, 734, 743, 752, 761, 770, 808, 817, 826, 835, 844, 853, 862, 871, 880, 909, 918, 927, 936, 945, 954, 963, 972, 981, 990, 1101, 2110, 2202, 3211, 3303, 4220, 4312, 4404, 5321, 5413, 5505, 6330, 6422, 6514, 6606, 7431, 7523, 7615, 7707, 8440, 8532, 8624, 8716, 8808, 9541, 9633, 9725, 9817, 9909, 21101, 32110, 42202, 53211, 63303, 64220, 74312, 84404, 85321, 95413, 96330, 321101, 532110, 642202, 853211, 963303

It can be noted that some numbers containing zero feature in both sequences. These numbers are 101, 202, 303, 404, 505, 606, 707, 808 and 909.

While we're at it why not consider multiplication in which the first two digits multiply together to give the third digit and so on. In the range up to one million, there are 78 such numbers and here they are in a sequence we'll call the MULTIPLICATION SEQUENCE (permalink):

100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 200, 212, 224, 236, 248, 300, 313, 326, 339, 400, 414, 428, 500, 515, 600, 616, 700, 717, 800, 818, 900, 919, 1000, 1111, 1224, 1339, 2000, 2122, 2248, 3000, 3133, 4000, 4144, 5000, 5155, 6000, 6166, 7000, 7177, 8000, 8188, 9000, 9199, 10000, 11111, 12248, 20000, 21224, 30000, 31339, 40000, 50000, 60000, 70000, 80000, 90000, 100000, 111111, 200000, 212248, 300000, 400000, 500000, 600000, 700000, 800000, 900000, 1000000

An example is 212248 where we have:$$2 \times 1 = 2, \, 2 \times 1 = 2, \, 2 \times 2 = 4 \text{ and } 4 \times 2 = 8$$The zeros of course make some of these numbers a little trivial and so with the digit 0 excluded we have 41 suitable numbers (permalink) in a sequence we'll call the MULTIPLICATION WITHOUT ZERO SEQUENCE:

111, 122, 133, 144, 155, 166, 177, 188, 199, 212, 224, 236, 248, 313, 326, 339, 414, 428, 515, 616, 717, 818, 919, 1111, 1224, 1339, 2122, 2248, 3133, 4144, 5155, 6166, 7177, 8188, 9199, 11111, 12248, 21224, 31339, 111111, 212248

If we consider dividing the second digit into the first to give the third digit and so on then, excluding numbers with zero, we have the following numbers (permalink) in what we'll call the DIVISION WITHOUT ZERO SEQUENCE:

111, 212, 221, 313, 331, 414, 422, 441, 515, 551, 616, 623, 632, 661, 717, 771, 818, 824, 842, 881, 919, 933, 991, 1111, 2212, 3313, 4221, 4414, 5515, 6616, 7717, 8422, 8818, 9331, 9919, 11111, 42212, 84221, 93313, 111111, 842212

Many of the numbers in the division sequence are not surprisingly the reverse of numbers in the multiplication sequence e.g. 842212 in the division sequence is the reverse of 212248 in the multiplication sequence.