For any integer b > 1, the baseb expansion of any number k < b will be a onedigit number, and will thus be trivially palindromic.
For each j >= 5 and odd, k = (j^3 + 6*j^2 + 14*j + 11)/2 is a term in the sequence, and represents a 3digit palindrome in each of three consecutive integer bases:
.
base 1st digit 2nd digit 3rd digit
   
j+1 (j+3)/2 (j+5)/2 (j+3)/2
j+2 (j+1)/2 (j+3)/2 (j+1)/2
j+3 (j1)/2 (j+7)/2 (j1)/2
.
(see 178 and 373 in the Example section). Nearly all of the first 95 terms of this sequence are terms of this form.
For each j >= 44 and divisible by 4, k = (3*j^5 + 30*j^4 + 125*j^3 + 270*j^2 + 307*j + 148)/4 is a term in the sequence, and represents a 5digit palindrome in each of three consecutive integer bases:
.
base 1st digit 2nd digit 3rd digit 4th digit 5th digit
     
j+1 3*j/4 + 4 j/2 + 9 j/4 + 11 j/2 + 9 3*j/4 + 4
j+2 3*j/4 + 1 j/2 + 2 j/4 + 0 j/2 + 2 3*j/4 + 1
j+3 3*j/4  2 j/2 + 10 j/4  11 j/2 + 10 3*j/4  2
.
[Reformatted by Jon E. Schoenfield, Apr 01 2018]
From Matej Veselovac, Mar 31 2018: (Start)
Similarly to the one 3digit and one 5digit families given above, at least seven more infinite families exist, for 7digit consecutive palindromes. Given a nonnegative integer n, we have the following representations palindromic in exactly three consecutive integer number bases j+1, j+2, j+3 :
1. For each j = 36+12n, k = (816 + 2474*j + 3114*j^2 + 2117*j^3 + 852*j^4 + 209*j^5 + 30*j^6 + 2*j^7)/12 is a term of the sequence.
2. For each j = 55+6n, k = (245 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/6 is a term of the sequence.
3. For each j = 73+2n, k = (247 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/2 is a term of the sequence.
4. For each j = 116+12n, k = (2440 + 7366 j + 9694 j^2 + 7171 j^3 + 3232 j^4 + 895 j^5 + 142 j^6 + 10 j^7)/12 is a term of the sequence.
5. For each j = 172+6n, k = (812 + 2446 j + 3290 j^2 + 2527 j^3 + 1190 j^4 + 343 j^5 + 56 j^6 + 4 j^7)/6 is a term of the sequence.
6. For each j = 288+12n, k = (1176 + 3566 j + 4374 j^2 + 2807 j^3 + 1032 j^4 + 227 j^5 + 30 j^6 + 2 j^7)/12 is a term of the sequence.
7. For each j = 277+6n, k = (1237 + 3740 j + 4900 j^2 + 3590 j^3 + 1600 j^4 + 440 j^5 + 70 j^6 + 5 j^7)/6 is a term of the sequence.
The smallest terms given by these families are of magnitudes ~ 10^10.3, 10^11.5, 10^12.8, 10^14.4, 10^15.5, 10^16.4 and 10^17. The smallest term of the next family, if it exists, is at least of magnitude ~ 10^18.
Almost all known terms of the sequence so far belong in one of the above defined families, either being 3, 5, or 7 digit palindromes in exactly 3 consecutive integer number bases.
There are 13 known terms that do not belong to any families: 300, 3360633, 19987816, 43443858, 532083314, 1778140759, 2721194733, 11325719295, 47622367425, 97638433343, 224678540182, 265282702996, 561091062285 (all but 300 so far are 7digit cases).
Infinite families for consecutive palindromes longer than 7 digits, as well as any examples for those cases, have not yet been observed.
Smallest example for 9digit consecutive palindromes does not exist within first 100 integer number bases, thus is at least > 10^16.
Similarly, no terms palindromic in 4 or more consecutive integer number bases have been found, so far.
[Extended by Matej Veselovac, Feb 05 2019] (End)
