A002113: Palindromes in base 10.

Initially, the sound is louder, has medium-loud pitch, and a subtle clicking noise in the background. The timbre varies from the rest of the sound, which is resonant and low-pitched. Here, the clicking is more noticeable. 

Residue counts:
2   [889, 1110]
3   [667, 666, 666]
4   [445, 555, 444, 555]
5   [223, 444, 444, 444, 444]
6   [295, 367, 295, 372, 299, 371]
7   [304, 281, 277, 280, 273, 292, 292]
8   [223, 278, 222, 277, 222, 277, 222, 278]
9   [223, 222, 222, 222, 222, 222, 222, 222, 222]
10   [1, 222, 222, 222, 222, 222, 222, 222, 222, 222]
11   [1090, 91, 91, 91, 91, 91, 91, 91, 91, 91, 90]
12   [148, 183, 146, 187, 148, 187, 147, 184, 149, 185, 151, 184]

Density information:
A(x) at x=10,100,1000,...:
[10, 19, 109, 199, 1099, 1999]

A(x)/x at x=10,100,1000,...:
1.0,0.19,0.109,0.0199,0.01099,0.001999,

First 100 first differences:
[1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 11, 11, 11, 11, 11, 11, 11, 2, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10]

First 99 second differences:
[0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,-9,8,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0]

Looking at the spectrogram, there are dark spectral lines at approximately 4386Hz and its multiples (x2, x3, x4). Dividing the sampling rate 44100 by 4386 is close to the integer 10. There are also less prominent spectral lines that happen at each multiple of 1/10th of ~4386Hz or 440Hz, and 44100/440 is about 100. The multiples of 440Hz are especially apparent in the peaks of the plot spectrum. The fact that 44100/4386 is approximately 10 is consistent with the distribution of residue counts for mod 10, which is nearly uniform with 222 terms for residues 1-9, except for 1 term for residue 0 because palindromes cannot end in 0 (since they cannot start with 0 except for 0 itself). In addition, the most common difference is 100 while the second common difference is 110. 

Moreover, the sequence is infinite. We can show this by considering the subsequence, 1, 11, 111, 1111,... Each term is a palindrome because it can be read the same forward and backward (and simply because every digit is 1). Since this subsequence is infinite, the sequence of base-10 palindromes is also infinite. 

The values of A(x)/x suggest that the density approaches 0, so we work to prove this. We let p(n) be the number of base-10 palindromes with n digits, and let A(x) be the number of palindromes <= x. If x = 10^k, then every palindrome less than 10^k has at most k digits, so A(10^k) = p(1) + p(2) +...+ p(k). To find an upper bound for p(n), we can look at how many digits are free to choose. A palindrome with n digits is determined by about the first half of its digits, and the rest are forced by symmetry. There are both even and odd cases for p(n), but both can be bounded above with 10^{(n+1)/2}. So we can write p(n) <= 10^{(n+1)/2}. Therefore, A(10^k) <= k*10^{(k+1)/2}. Dividing by 10^k, we get A(10^k)/10^k <= (k*10^{(k+1)/2})/10^k. (k*10^{(k+1)/2})/10^k approaches 0 as k goes to infinity because 10^k grows faster than k*10^{(k+1)/2}. The proportion of integers up to 10^k that are palindromes approaches 0. Thus, the density approaches 0.

-- Lauren Yee