Can a natural number be nontrivially palindromic in more than $3$ consecutive integer bases?
Nontrivially means that I'm not counting one-digit palindromes.
The initial question was asked on Math.SE - but was not solved and still isn't.
Examples for exactly $3$ consecutive bases are mentioned in this OEIS sequence.
Update: (added patterns from initial post, and some more on them)
I believe it can be proved that a solution for $4$ or more consecutive bases does not exist (if this is the case), if all patterns for $3$ consecutive bases are found, since at that point, it would be possible to show if the patters will or never will extend to a fourth consecutive base. (As was shown in the comments and then the same claim in the answer of the Math.SE post for 3 digit pattern)
Even length palindromes seem to not form consecutive palindromes for three or more bases - still don't know how to show why this is true.
Following that, then we choose to observe odd digit length palindromes of $2d+1$ digits, $d\in\mathbb N$, and we have so far:
$$(d=1)$$
All examples for this case are of form: ($n=2k+3, k\in\mathbb N$)
$$\frac{1}{2}(n^3 + 6n^2 + 14n + 11)$$
And palindromic in bases $n+1, n+2, n+3$; Except number $300 = 606_7 = 454_8 = 363_9$
How to show that this is the only pattern and only exception for this case?
(Verified up to base $b$; No new exceptions or patterns exist for $b\lt1400$, so far)
$$(d=2)$$
All examples for this case are of form: ($n=4k+40, k\in\mathbb N$)
$$\frac{1}{4}(3n^5 + 30n^4 + 125n^3 + 270n^2 + 307n + 148)$$
And palindromic in bases $n+1, n+2, n+3$; No Exceptions found.
How to show that this is the only pattern and that there are no exceptions?
(No new exceptions or patterns exist for any numbers in all bases $b\lt220$, so far)
$$(d=3)$$
For $7$-digit palindromes, I do not see a clear pattern. All the examples so far computed below:
(for $b\le78$, using this python code), Read columns: [base, decimal value, digits in base]
9 3360633 7
13 19987816 7
15 43443858 7
22 532083314 7
26 1778140759 7
28 2721194733 7
28 11325719295 7
36 47622367425 7
37 19683596522 7
40 97638433343 7
42 224678540182 7
43 265282702996 7
48 561091062285 7
49 133256051308 7
56 326217315210 7
61 597702412638 7
62 657158314249 7
68 1242101453540 7
73 2055729074336 7
74 2226313335987 7
74 6678940007962 7
76 8029674745361 7
78 9608108112996 7
These are palindromic in column given base $b$ and bases $b+1,b+2$.
Can a pattern be found here, or anything that can relate and generate these examples so far?
$$(d=4)$$
No examples have been found so far which isn't surprising as I only checked all palindromes in number bases $b\lt39$, so far, and the first example for $d=2$ shows at base $b=45$ as seen above, for comparison. Checking all the bases since $d=3$ surprisingly has first example at $b=9$ as seen above.
But this took five hours with the linked python code to check; where the base $b=38$ alone took one hour. (checking every single nine digit plindrome); Pastebin link for time comparison.
$$(d\ge5)$$
Haven't searched for examples, as $d=4$ is already taking a long time per number base.
You can see in the linked code that I'm checking each palindrome, but if you look at examples for $d=3$, they are almost always increasing, thus a lower bound would speed up the search.