A positive whole number that is the same when reading from right to left and left to right is called a palindromic number. If all of them were put into a sequence that is from small to big:1,2...11,22...,101,111...,10001,10101,.....

Then which term is the number 78987 in this sequence? For example : 1 is the 1st term,2 is the second....etc

yomyhomies
Oct 22, 2017

#2**+1 **

Starting with 0, 78987 is the 889th palindrome

[BTW...889 is known as the palindrome's* rank* ]

Starting with 1, it is the 888th palindrome

This can be verified with the calculator here :

http://rhyscitlema.com/algorithms/generating-palindromic-numbers/

[ The calculator is about half-way down the page.....there are also other intems of interest related to palindromes on this page ]

A procedure is desribed for finding the nth palindrome, but I don't see one that tells us how to determine the rank of any particular palindrome [ although I did not look at the page with a fine tooth comb]

Maybe heureka knows of a proceedure to produce this???

CPhill
Oct 22, 2017

#3**+3 **

**which term is the number 78987 in this sequence?**

Here 0 is the 1st term:

The position of a **palindrome **within the sequence can be determined almost without calculation:

If the **palindrome **has an **even number of digits**,

prepend a 1 to the front half of the** palindrome's** digits.

Examples: 98766789=a(19876)

If the **number of digits is odd**, prepend the value of front digit + 1 to the digits from position 2 ... central digit.

Examples: 515=a(61), 8206028=a(9206), 9230329=a(10230).

see link: http://oeis.org/search?q=Palindromes+in+base+10

which term is the number **78987** in this sequence?

The **number of digits ****is 5**** is odd**, prepend the value of front digit + 1 to the digits from position 2 ... central digit.

\(\begin{array}{|rrrrll|} \hline & & & \Rsh & & \text{ until central digit} \\ &7 & 8 & 9 & 8 & 7 \\ &| & | & | \\ &+1 & | & | \\ &\downarrow & \downarrow & \downarrow \\ \text{term is } & \color{red}8 &\color{red}8 &\color{red} 9 \\ \hline \end{array} \)

Starting with 1, it is the **888th** palindrome

heureka
Oct 23, 2017