Categories of Palindromic Numbers
No. of Digits  Prime  Triangular  Perfect square  sum of consecutive squares (2 or more)  cube  product of successive numbers 
1  1,2,3,5,7  1,3,6  1,4,9  5  1  
2  11  55,66    55,77  
3  101,131,151,181, 191,313,353,373, 383,727,757,787, 797.919,929 
171,595,666  121,484,676  121,181,313,434,484,505,545,595,636,676,818,  343  272=16*17 
4  Nil  3003,5995.8778    1001,1111,1441,1771,4334,6446,  1331  6006=77*78 
5  93 primes  15051,66066  10201,12321,14641,40804 44944,69696,94249

10201,12321,14641,17371,17871,19691,21712, 40804,41214,42924,44444,44944,46564,51015, 65756,69696,81818,94249,97679,99199 

6  Nil  617716,828828  698896  289982=538*539  
7  668 primes  1269621,1680861 3544453,5073705 5676765,6295926 
1002001,1234321,4008004, 5221225,6948496, 
1030301(101^{3})  6039306=2457*2458 
Properties of Palindromic Numbers 
*Palindromes have their origin from the Greek word palindromos meaning "running back again". 
* Take any number. Add it to the reverse number. If it it is not a palindrome, again add the new number to its reverse till it becomes a palindrome. 80% of the numbers under 10000 yield palindromes in 4 or less steps. about 10% in more than 4 but up to 7 steps. A rare number 89 as well as 98 take 24 iterations to become a palindrome. Palindrome words  eye, noon, radar, mom, dad, deed, civic, level 
* There are some numbers which do not form a palindrome in the above process. These are called Lychrels. The first Lychrels is 196. The reverse no. 691 is also Lychrels. If one goes on adding these numbers as above, all the numbers generated are lychrels. For example, in the chain of (196,691)>(788,887), (1675, 5761) and up above are all lychrels. 
* All palindromes with even number of digits are divisible by 11.Therefore barring 11 which is a 2 digit Palindrome, there are no palindromes with 4,6, 8, ..... no. of digits. All even digit palindromes can be expressed as 100a( 10^{1} + 1) + 10b (10^{3} + 1) + c (10^{5} +1) +.... ( this is the format for 6 digit palindrome. One can construct on similar lines for more or less number of even digits ) & each coefficient reduces to the equation 10 ^{ (2k+1)} +1 = 11x where k, x have integer values. 
* All Palindromic numbers which are triangular must have either of 0,1,3,5,6,8 as their last digit and the lower of the product digits of the triangular number has to be even. In other words for n(n+1)/2 to be palindromic, n is to an even number. 
* In molecular biology, the DNA and RNA sequences that read the same from both the ends i.e 5' & 3' end are called Palindromes. The sites of many restriction enzymes such as Restriction Endonuclease that cut DNA are palindromes. Long DNA palindromes pose a threat to genome stability. This instability is primarily mediated by slippage on the lagging strand of the replication fork between short directly repeated sequences close to the ends of the palindrome. The role of the palindrome is likely to be the juxtaposition of the directly repeated sequences by intrastrand basepairing. This intrastrand basepairing, if present on both strands, results in a cruciform structure. In bacteria, cruciform structures have proved difficult to detect in vivo, suggesting that if they form, they are either not replicated or are destroyed. SbcCD, a recently discovered exonuclease of Escherichia coli, is responsible for preventing the replication of long palindromes. These observations lead to the proposal that cells may have evolved a postreplicative mechanism for the elimination and/or repair of large DNA secondary structures It has been found that Direct and inverted repeats elicit genetic instability by both exploiting and eluding DNA doublestrand break repair systems in mycobacteria. It has also been found that widespread and nonrandom distribution of DNA palindromes in cancer cells provides a structural platform for subsequent gene amplification 
* Frequency of Palindrome distribution :
(n1) ^{k/2} where k is even and
(n1) ^{(k+1)/2} where k is odd.
n represents the Base system and k is the total number of digits in the
number. * In general for a number in any base n with k digits the number
of unique reverses of k digits is given by * General formulae which give the frequency of palindromes for all numbers in any base n less than or equal to k digits. If k is even, the formula = 2(n  l)[(n  1)^{k/2}  1]
If k is odd, formula = 2(n  l)[(n  1)^{(k+1)/2}  1]
 (n  1)^{(k+1)/2} 
References  Palindromic Numbers 
Mathematischa 