Find the number of positive integer palindromes written in base 10, with no zero digits, and whose digits add up to 13. For example, 42124 has these properties. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left.
Hint 3. n=3 aba⇒2a+b=13a1234567b11×9✓7✓5✓3✓1✓−1×⟹5
Hint 4. n=4 abba⇒2a+2b=13⇒a+b=213×⇒n should be odd
Hint 5. n=5 a1a2a3a2a12a1+2a2+a3=13a1,a2,a3≥1⇒a3 is odd Let a3=2a3′−1, a3′≥1. 2a1+2a2+2a3′−1=13a1+a2+a3′=7 By Stars and Bars, (26)=15 cases
Hint 6. n=7 a1a2a3a4a3a2a1 Similarly, 2a1+2a2+2a3+a4=13 Let a4=2a4′−1. a1+a2+a3+a4′=7a1,a2,a3,a4′≥1⇒(36)
Hint 7. Similarly n=9⇒(46)=15n=11⇒(56)=6n=12⇒(66)=1
Hint 8. By Hints 3, 5, 6, and 7 and Absolute Value, Ans=5+(26)+(36)+(46)+(56)+(66)=−2+((06)+(16)+(26)+⋯+(66))=−2+26=−2+64=62