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Lets look at sequences of numbers that are a palindromic number in two consecutive number bases $b$ and $b+1$, where $b\ge2$ of course. (And also ignoring trivial one digit palindromes.)

I would conjecture that there are infinitely many numbers palindromic in two consecutive number bases for any two number bases $(b, b+1)$ where $b\ge2$. But I do not know how to show that this statement is true.

Turns out, it is in fact not known if the case $(2,3)$ for example, has infinitely many terms, since the OEIS entry for it is written as "...if it exists". There are no clear patterns in this particular sequence, as it seems.

Numbers up to $10^7$ in number bases up to $32$ (where the $*$ indicates that the number is also palindromic in a third consecutive base) :

(2, 3): 6643, 1422773, 5415589, 
(3, 4): 10, 130, 11950, 175850, 749470, 1181729, 
(4, 5): 46, 9222, 76449, 193662, 2347506, 2593206, 
(5, 6): 67, 98, 104, 651, 2293, 3074, 26691, 27741, 704396, 723296, 755846, 883407, 
(6, 7): 92, 135, *178, 185, 5854, 6148, 7703, 186621, 204856, 206620, 213970, 269957, 271721, 279071, 4252232, 
(7, 8): 121, 178, 235, 292, *300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, 450985, 502457, 626778, 786435, 
(8, 9): 154, 227, 300, *373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, 1257716, 1262900, 1268084, 1273268, 1294652, 1368461, 1373645, 1405397, 2067519, 
(9, 10): 191, 282, 373, 464, 555, 646, 656, 6886, 25752, 27472, 42324, 50605, 626626, 1540451, 1713171, 1721271, 1828281, 1877781, 1885881, 2401042, 2434342, 2442442, 2450542, 3106013, 3114113, 3122213, 3163613, 3171713, 3303033, *3360633, 
(10, 11): 232, 343, 454, 565, *676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854, 4593954, 5643465, 5655565, 5667665, 5741475, 7280827, 7292927, 8710178, 8722278, 8734378, 8746478, 8758578, 8820288, 8832388, 8844488, 
(11, 12): 277, 410, 543, 676, 809, 942, 1075, 1208, 1220, 38425, 54662, 72351, 75399, 93088, 125430, 1798303, 1817179, 5058385, 5075809, 5093233, 5199361, 5216785, 5550889, 5568313, 5585737, 5603161, 5620585, 7569434, 7727702, 7833830, 7851254, 7868678, 7886102, 9711399, 9728823, 9746247, 
(12, 13): 326, 483, 640, 797, 954, *1111, 1268, 1425, 1582, 1595, 53210, 100636, 104549, 123257, 129198, 151819, 174596, 227806, 8281118, 8305454, 8329790, 8354126, 8502170, 8526506, 9041475, 9065811, 9090147, 9114483, 
(13, 14): 379, 562, 745, 928, 1111, 1294, 1477, 1660, 1843, 2026, 2040, 71905, 105394, 136517, 167458, 170006, 174934, 205875, 208423, 239364, 270487, 342392, 344954, 
(14, 15): 436, 647, 858, 1069, 1280, 1491, *1702, 1913, 2124, 2335, 2546, 2561, 95146, 139667, 181248, 225769, 231874, 267140, 276395, 317766, 454493, 499014, 502179, 
(15, 16): 497, 738, 979, 1220, 1461, 1702, 1943, 2184, 2425, 2666, 2907, 3148, 3164, 123617, 181698, 294260, 348501, 359797, 414038, 472119, 526600, 650217, 708298, 712154, 
(16, 17): 562, 835, 1108, 1381, 1654, 1927, 2200, *2473, 2746, 3019, 3292, 3565, 3838, 3855, 158050, 232579, 307108, 377285, 447190, 451814, 460807, 521719, 530712, 535336, 605241, 679770, 833468, 907997, 977902, 982526, 987167, 
(17, 18): 631, 938, 1245, 1552, 1859, 2166, 2473, 2780, 3087, 3394, 3701, 4008, 4315, 4622, 4640, 199225, 293474, 387723, 476770, 571019, 659760, 675996, 764737, 858986, 1147258, 1241507, 1330248, 1341282, 
(18, 19): 704, 1047, 1390, 1733, 2076, 2419, 2762, 3105, *3448, 3791, 4134, 4477, 4820, 5163, 5506, 5525, 247970, 365619, 483268, 712410, 823561, 842732, 941210, 953883, 1071532, 1189181, 1430995, 1548644, 1666293, 1777444, 
(19, 20): 781, 1162, 1543, 1924, 2305, 2686, 3067, 3448, 3829, 4210, 4591, 4972, 5353, 5734, 6115, 6496, 6516, 305161, 450322, 595483, 740644, 878585, 1016146, 1023746, 1161307, 1176147, 1183747, 1321308, 1466469, 1611630, 1909571, 2054732, 2192293, 2199893, 2337454, 2352294, 
(20, 21): 862, 1283, 1704, 2125, 2546, 2967, 3388, 3809, 4230, *4651, 5072, 5493, 5914, 6335, 6756, 7177, 7598, 7619, 371722, 548963, 726204, 903445, 1072286, 1249527, 1417948, 1444009, 1595189, 1612430, 1789671, 1966912, 2330234, 2507475, 2684716, 2853137, 3030378, 3047619, 
(21, 22): 947, 1410, 1873, 2336, 2799, 3262, 3725, 4188, 4651, 5114, 5577, 6040, 6503, 6966, 7429, 7892, 8355, 8818, 8840, 448625, 662994, 877363, 1091732, 1510768, 1714973, 1929342, 1949230, 2163599, 2377968, 2592337, 3031260, 3245629, 3459998, 3664203, 3878572, 3898460, 
(22, 23): 1036, 1543, 2050, 2557, 3064, 3571, 4078, 4585, 5092, 5599, *6106, 6613, 7120, 7627, 8134, 8641, 9148, 9655, 10162, 10185, 536890, 793939, 1050988, 1308037, 1565086, 1811003, 2056414, 2068052, 2313463, 2347894, 2570512, 2593305, 2850354, 3107403, 3364452, 3633161, 3890210, 4147259, 4392670, 4404308, 4649719, 4906768, 4929561, 
(23, 24): 1129, 1682, 2235, 2788, 3341, 3894, 4447, 5000, 5553, 6106, 6659, 7212, 7765, 8318, 8871, 9424, 9977, 10530, 11083, 11636, 11660, 637585, 943394, 1249203, 1555012, 1860821, 2153934, 2459743, 2752304, 3058113, 3084081, 3389890, 3695699, 4001508, 4626397, 4932206, 5238015, 5530576, 5836385, 6142194, 6168162, 
(24, 25): 1226, 1827, 2428, 3029, 3630, 4231, 4832, 5433, 6034, 6635, 7236, *7837, 8438, 9039, 9640, 10241, 10842, 11443, 12044, 12645, 13246, 13271, 751826, 1113027, 1474228, 1835429, 2196630, 2904632, 3250833, 3612034, 3973235, 4002660, 4363861, 4725062, 5086263, 5462488, 5823689, 6184890, 6546091, 6892292, 7253493, 7614694, 7644119, 
(25, 26): 1327, 1978, 2629, 3280, 3931, 4582, 5233, 5884, 6535, 7186, 7837, 8488, 9139, 9790, 10441, 11092, 11743, 12394, 13045, 13696, 14347, 14998, 15024, 880777, 1304578, 1728379, 2152180, 2575981, 2999782, 3407333, 3814234, 3831134, 4238035, 4661836, 4695012, 5118813, 5542614, 5966415, 6830942, 7254743, 7678544, 8085445, 8102345, 8509246, 8933047, 9356848, 9390024, 
(26, 27): 1432, 2135, 2838, 3541, 4244, 4947, 5650, 6353, 7056, 7759, 8462, 9165, *9868, 10571, 11274, 11977, 12680, 13383, 14086, 14789, 15492, 16195, 16898, 16925, 1025650, 1519859, 2014068, 2508277, 3002486, 3496695, 3972652, 4466861, 4942116, 5436325, 5930534, 5967767, 6461976, 6956185, 7450394, 7963583, 8457792, 8952001, 9446210, 9921465, 
(27, 28): 1541, 2298, 3055, 3812, 4569, 5326, 6083, 6840, 7597, 8354, 9111, 9868, 10625, 11382, 12139, 12896, 13653, 14410, 15167, 15924, 16681, 17438, 18195, 18952, 18980, 1187705, 1760754, 2333803, 2906852, 3479901, 4052950, 5178636, 5730517, 6303566, 6876615, 6918223, 7491272, 8064321, 8637370, 9804663, 
(28, 29): 1654, 2467, 3280, 4093, 4906, 5719, 6532, 7345, 8158, 8971, 9784, 10597, 11410, *12223, 13036, 13849, 14662, 15475, 16288, 17101, 17914, 18727, 19540, 20353, 21166, 21195, 1368250, 2029219, 2690188, 3351157, 4012126, 4673095, 5334064, 5972297, 6609718, 6633266, 7270687, 7931656, 8592625, 8638938, 9299907, 9960876, 
(29, 30): 1771, 2642, 3513, 4384, 5255, 6126, 6997, 7868, 8739, 9610, 10481, 11352, 12223, 13094, 13965, 14836, 15707, 16578, 17449, 18320, 19191, 20062, 20933, 21804, 22675, 23546, 23576, 1568641, 2327282, 3085923, 3844564, 4603205, 5361846, 6120487, 6853898, 7612539, 8345080, 9103721, 9862362, 9913722, 
(30, 31): 1892, 2823, 3754, 4685, 5616, 6547, 7478, 8409, 9340, 10271, 11202, 12133, 13064, 13995, *14926, 15857, 16788, 17719, 18650, 19581, 20512, 21443, 22374, 23305, 24236, 25167, 26098, 26129, 1790282, 2657043, 3523804, 4390565, 5257326, 6124087, 6990848, 8696470, 9534401, 
(31, 32): 2017, 3010, 4003, 4996, 5989, 6982, 7975, 8968, 9961, 10954, 11947, 12940, 13933, 14926, 15919, 16912, 17905, 18898, 19891, 20884, 21877, 22870, 23863, 24856, 25849, 26842, 27835, 28828, 28860, 2034625, 3020674, 4006723, 4992772, 5978821, 6964870, 7950919, 8936968, 9892265, 
(32, 33): 2146, 3203, 4260, 5317, 6374, 7431, 8488, 9545, 10602, 11659, 12716, 13773, 14830, 15887, 16944, *18001, 19058, 20115, 21172, 22229, 23286, 24343, 25400, 26457, 27514, 28571, 29628, 30685, 31742, 31775, 2303170, 3420419, 4537668, 5654917, 6772166, 7889415, 9006664, 

Looks like the first $9$ sequences for these are in the OEIS.
OEIS links: $(2,3)$, $(3,4)$, $(4,5)$, $(5,6)$, $(6,7)$, $(7,8)$, $(8,9)$, $(9,10)$, $(10,11)$.

Notice that I'm excluding one digit palindromes in my intended sequences, which are included in the OEIS links, for comparison.



I noticed patterns in the terms of these sequences, as shown below.

All the sequences $(b,b+1)$ follow the formulas (patterns):


$$f_1(n)=((b+1)^2-b)n+(b+1)^2$$

This gives the $n$th term, but this is only valid for first $(b-3)$ terms.
This pattern appears for $b\gt3$.


Then the $(b-2)$th term equals $f_1(b-2)-b^2$, for $b\gt4$.


Then there is a second pattern, following the formula:

$$f_2(n)=((b+1)^4-b(2b^2+3b+2))n+(b+1)^4$$

Were $n=1$ at $(b-1)$ and increasing by $1$ normally, up to the term $(b+\lceil\frac{b}{3}\rceil-5)$.
This pattern appears for $b\gt9$.


Then the term after that, the $(b+\lceil\frac{b}{3}\rceil-4)$ equals... ?



With these patterns so far, we can calculate:

  • first $(b-3)$ terms for $b\gt3$
  • first $(b-2)$ terms for $b\gt4$
  • first $(b+\lceil\frac{b}{3}\rceil-5)$ terms for $b\gt9$

Observation:

If you decrease $b$ by $1$ in the pattern functions, and expand the expressions, we get:

$$f_1=(1b^2-1b+1)n+b^2$$ $$f_2=(1b^4-2b^3+3b^2-2b+1)n+b^4$$

You can see that the coefficients next to $b$ on the left, make $111$ and $12321=111^2$;

Question

What is the next pattern expressible in form of a polynomial function to extend the calculations to up to first X terms, for $b$ greater than Y?

How can we obtain all such patterns? Can these patterns be generalized into a single formula?

I wouldn't be surprised if it follows $111^x$, but why is that? Can we reach this without checking each pattern against the calculated data? And determine the bounds for terms it is valid?

If we found all such patterns, notice that sequence $(2,3)$ for example, will never have any of these patterns. The same as the bases $b\le9$ will not contain the next pattern that is yet to be found, since the Y can only grow as we acquire the next pattern, if I'm not mistaken.


This begs a follow up question;

Can we find any possible patterns for values that are not included in the patterns of these kind?



Remarks

Note that you could define a function $f$ that generates palindromes in base $b$ and then define a function $g$ that is an intersection of $f(b)$ and $f(b+1)$ to get $(b,b+1)$; but I'm not looking to define or generate these sequences, rather; In finding computable sequence formulas like these patterns so far.

These patterns so far look like they should have an obvious generalization and extension, but I'm not seeing it unless I went on to check more data for each individual pattern hoping to confirm the observed pattern in the patterns.

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