A002113: Palindromes in base 10. (10^6 - 2 * 10^6)

The sound is different than the one generated by the terms up until 10^6. It is constant with the same medium-low tone and volume. Slightly clickly.

Residue counts:
2   [0, 1000]
3   [333, 333, 334]
4   [0, 500, 0, 500]
5   [0, 1000, 0, 0, 0]
6   [0, 333, 0, 333, 0, 334]
7   [145, 146, 145, 142, 140, 140, 142]
8   [0, 250, 0, 250, 0, 250, 0, 250]
9   [111, 111, 112, 111, 111, 111, 111, 111, 111]
10   [0, 1000, 0, 0, 0, 0, 0, 0, 0, 0]
11   [91, 91, 91, 90, 91, 91, 91, 91, 91, 91, 91]
12   [0, 166, 0, 166, 0, 167, 0, 167, 0, 167, 0, 167]

Density information:
A(x) at x=10,100,1000,...:
[0, 1, 2, 3, 4, 5, 1000]

A(x)/x at x=10,100,1000,...:
0.0,0.01,0.002,0.0003,4e-05,5e-06,0.0001,

First 100 first differences:
[1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1100, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 110]

First 99 second differences:
[0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,100,-100,0,0,0,0,0,0,0,0,-890]

The previous spectrogram had dark spectral lines at approximately 4386Hz and its multiples (x2, x3, x4), whereas this spectrogram has no prominent spectral lines. From the plot spectrum, there are evenly spaced peaks across the frequency range. The sound remains relatively constant, with uniformly repeating clicks. This behavior makes sense from the sequence values and their positional symmetry. In palindromes such as 1000001, 1001001, 1002001, etc., the first and last digits must match, the second and second-to-last digits must match, and the third and third-to-last digits must match. As a result, changing one digit automatically determines another digit symmetrically.

From 0 to 10^6, the gaps are mostly 100, whereas from 10^6 to 2*10^6, the gaps are dominated by repeated values of 1000. The palindromes become more spread out from 10^6 to 2*10^6 because 7-digit palindromes are determined by the first four digits and changing one of those digits often changes the value by around 1000 instead of 100. We can prove this by writing a 7-digit palindrome as 1abcba1 where a, b, c can each be from 0 to 9. We can break down the palindrome like so:
1abcba1 = 1000001 + 100000(a) + 10000(b) + 1000(c) + 100(b) + 10(a)
	= 1000001 + 100010(a) + 10100(b) + 1000(c)

The gaps are mostly 1000 because c is the digit that changes most frequently. For fixed a and b, the palindromes run through c = 0,1,..,9 and each increase in c raises the value by 1000. This creates 9 gaps of 1000 for every fixed a and b. Only when c resets from 9 back to 0 does b increase, and when both b and c reset, a increases. These changes create the less frequent gaps, such as 1100.

With the first few A(x)/x terms, it seems that the density decreases quickly as x increases. The values become extremely small, suggesting that the sequence becomes increasingly sparse with larger integers. We can be more precise about this by counting palindromes based on their digit structure. For a 3-digit palindrome of the form xyz, the first digit x has 9 choices (1–9), the middle digit y has 10 choices, and then, z has 1 choice because it must equal x. Therefore, there are 9 × 10 = 90 three-digit palindromes. More generally, for an n-digit palindrome, only about half of the digits can vary freely because the remaining digits are determined by positional symmetry. If n is odd, p(n) = 9 * 10^((n−1)/2). If n is even, p(n) = 9 * 10^(n/2 − 1).

In either case, p(n) grows on the order of 10^(n/2), while the total number of n-digit integers grows on the order of 10^n. Therefore, the ratio p(n)/10^n decreases toward 0 as n increases. This explains why A(x)/x becomes smaller and why palindromes become increasingly sparse among larger integers.

-- Lauren Yee