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.