[ January 31, 2024 ] [ Last update – ]
Detecting patterns in sum of the first n palindromes.
By Tony Sand (email)

Apply the method '\(\color{blueviolet}{\text{recording the indices when the terms of a sequence increases}}\)
\(\color{blueviolet}{\text{in length (base_10) by one digit compared to its previous term}}\)' to the sequence
of the triangular numbers (A000217) with formula (k^2+k)/2.
Very quickly you'll notice that the new sequence on the right side is a twofold
alternation of the approximations to the digits of sqrt(2) and sqrt(20).
\(\sqrt{2} = 1,41421356\ldots\) and \(\sqrt{20} = 4,472135955\ldots\)

\(\bbox[8px,border:1px yellow solid]{ \begin{align} 1 &\to 1\\ 10 &\to 4\\ 105 &\to 14\\ 1035 &\to 45\\ 10011 &\to 141\\ 100128 &\to 447\\ 1000405 &\to 1414\\ 10001628 &\to 4472\\ 100005153 &\to 14142\\ 1000006281 &\to 44721\\ 10000020331 &\to 141421\\ 100000404505 &\to 447214\to \scriptsize{\sqrt{20}}\\ 1000001326005 &\to 1414214\to \scriptsize{\sqrt{2}} \end{align} }\)

Next I was applying this method to the sequence A046489
titled 'Sum of the first n palindromes' and surprisingly a threefold pattern emerged
as illustrated with the three colors.

\(\bbox[8px,border:1px maroon solid]{ \begin{align} 1 &\to 1\\ 10 &\to 4 \\ 111 &\to 12\\ \\ 1004 &\to 22\\ 10070 &\to 53\\ 102642 &\to 132\\ \\ 1009749 &\to 237\\ 10033308 &\to 545\\ 100154396 &\to 1330\\ \\ 1001369313 &\to 2381\\ 10004196212 &\to 5461\\ 100024288068 &\to 13314\\ \\ 1000006759460 &\to 23818\\ 10000212832964 &\to 54619\\ 100001115019389 &\to 133151\\ \\ 1000007042189240 &\to \color{red}{238198}\\ 10000034762413704 &\to \color{green}{546204}\\ 100000120903813131 &\to \color{blue}{1331525}\\ \\ 1000000135026527885 &\to 2381993\\ 10000003532393213172 &\to 5462051\\ 100000021451421850551 &\to 13315265\\ \\ 1000000135057109873490 &\to 23819948\\ 10000000409012039207852 &\to 54620521\\ 100000001559889033349352 &\to 133152662\\ \\ 1000000010677994338403061 &\to 238199489\\ 10000000007427979784149364 &\to 546205219\\ 100000000233783283781587628 &\to 1331526634\\ \\ 1000000001004034700313682433 &\to 2381994901\\ 10000000002965934808413585636 &\to 5462052207\\ 100000000001714647986654094989 &\to 13315266351 \end{align} }\)

Alas, after trying various roots and powers I didn't came accross approximations
for these three terms. I don't understand why? Can you draw on the wisdom of crowds?
I'm sure there must be an answer hidden somewhere.