Can a natural number be nontrivially palindromic in more than $3$ consecutive number bases?

Nontrivially means that I'm not counting one-digit palindromes.

Was asked on MSE - but wasn't solved and went inactive, even after a bounty.

$$\text{Looking at consecutive palindromes in three bases}$$

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.

(Like it was shown on MSE for the 3 digit pattern) - This still does not exclude the 3 digit palindromes, since it is not proven if this is the only pattern. See more under $d=1$ below.

$(0)$ Even length palindromes ($2p$ digits, $p\in\mathbb N$) seem to not form consecutive palindromes in three or more bases - still don't know how to prove this observation.

Following that, then we choose to observe odd digit length palindromes of $2d+1$ digits, $d\in\mathbb N$, which are palindromic in $3$ consecutive number bases $b\in\mathbb N \ge 2$, and we have so far:

Pick ($n=2k+3, k\in\mathbb N$), then all examples for this case are of form:

$$\frac{1}{2}(n^3 + 6n^2 + 14n + 11)$$

And palindromic in bases $n+1, n+2, n+3$; Except number $300 = 606_7 = 454_8 = 363_9$

$(1)$ How to show (prove) that all 3 digit examples other than $300$ are in this pattern?

(No new exceptions or patterns exist for $b\lt2333$, so far. Verified here using python code)

Pick ($n=4k+40, k\in\mathbb N$), then all examples for this case are of form:

$$\frac{1}{4}(3n^5 + 30n^4 + 125n^3 + 270n^2 + 307n + 148)$$

And palindromic in bases $n+1, n+2, n+3$; No Exceptions found.

$(2)$ How to show (prove) that all 5 digit examples are in this pattern?

(No exceptions or new patterns exist for $b\le333$, so far. Verified here using python code)

Update: Based on computed examples and observations so far, multiple patterns and exceptions seem to exist for this case. First pattern that occurs:

Pick ($n=2k+79, k\in\mathbb N\cup \{-3,-2,-1\}$), then there are infinitely many examples of form:

$$\frac{1}{2}(n^7+14n^6+88n^5 + 320n^4 + 718n^3 + 980n^2 + 748n + 247)$$

And are palindromic in number bases $n+1, n+2, n+3$.

Numbers that seem to form the second pattern: (And are palindromic in $(b), (b+1), (b+2)$)

326217315210 (56), 
657158314249 (62), 
2226313335987 (74), 
3815123088334 (80), 
6290902501325 (86), 
10032985497864 (92), 
15540762075415 (98), 
23460181868882 (104), 
... (106), etc.

Numbers that seem to form the third pattern: (And are palindromic in $(b), (b+1), (b+2)$)

19987816 (13), 
19683596522 (37), 
133256051308 (49), 
597702412638 (61), 
2055729074336 (73), 
5872897399570 (85), 
14629218708372 (97),
... (109), etc.

Numbers that seem to be exceptions, and the only exceptions (not belonging to any patterns):

9 3360633 
15 43443858 
22 532083314 
26 1778140759 
28 2721194733 
28 11325719295 
36 47622367425 
40 97638433343 
42 224678540182 
43 265282702996 
48 561091062285 
68 1242101453540 

These are palindromic in column given base $b$ and bases $b+1,b+2$.

(3) I believe the second and third patterns belong to a similar polynomial formula as the first pattern. I will find and post them, but first I need to compute more terms. I also believe no more exceptions than these listed, exist. These findings of course, also need to be proven.

(Exceptions and patterns verified for all $b\lt110$, so far. Computed here using python code)

No examples have been found so far. I've checked number bases $b\lt50$ so far, here.

Haven't searched for examples yet, as $d=4$ is already taking a long time per number base.

Some of these examples and patterns are also mentioned in an OEIS sequence.

I will keep updating the post as new examples are computed.

Can the python code I'm using be more optimized? (Is there a faster way to compute this?)

How can one find these polynomial pattern expressions algebraically? Is there anything out there that can actually attack this problem?