Palindromic Sums

The palindromic number $595$ is interesting because it can be written as the sum of consecutive squares: $6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2$.
There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is $4164$. Note that $1 = 0^2 + 1^2$ has not been included as this problem is concerned with the squares of positive integers.
Find the sum of all the numbers less than $10^8$ that are both palindromic and can be written as the sum of consecutive squares.

This problem requires knowledge of both the properties of palindromic numbers and of sums of squares. Palindromic numbers are numbers that read the same backward as forward. The sum of consecutive squares is a specific form of arithmetic sequence, a sequence of numbers in which the difference between consecutive terms is constant.

The strategy to solve this problem in an efficient way is to generate all palindromic numbers less than $10^8$ and check if they can be expressed as the sum of consecutive square numbers.

However, it’s the kind of problem that can’t be solved analytically. You need to apply programming concepts to solve this problem and its complexity exceeds the scope of what can be done manually.

To begin solving, you would need to create an algorithm that does the following:

1. Generates all palindromic numbers less than $10^8$. There are a couple of different strategies to generate them. You could generate all numbers less than $10^8$ and check if they are palindromic or you could directly generate the palindromic numbers.

2. After generating a palindromic number, the algorithm has to check if it can be expressed as the sum of consecutive square numbers. Here you can make use of the identity $(n+a)^2 – n^2 = 2an + a^2$. If a number can be expressed as the sum of squares, it will be the sum of a series of terms of the form $2an+a^2$ for $a=0,1,…,$ where $2n+1$ is a square.

3. Finally, the program has to keep track of the palindromic numbers that can also be expressed as the sum of consecutive squares and return their sum.

This would be best done using a programming language like Python, C++, or Java, where you could efficiently generate and check these numbers. It can’t be manually calculated because it involves dealing with large numbers and it would be time-consuming to do it manually.

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