Palindromic Triangular Numbers with Palindromic Index

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Sequence

The palindromic triangular numbers whose indices are themselves palindromic are:

\(\displaystyle T_1\) \(=\) \(\displaystyle 1\)
\(\displaystyle T_2\) \(=\) \(\displaystyle 3\)
\(\displaystyle T_3\) \(=\) \(\displaystyle 6\)
\(\displaystyle T_{11}\) \(=\) \(\displaystyle 66\)
\(\displaystyle T_{77}\) \(=\) \(\displaystyle 3003\)
\(\displaystyle T_{363}\) \(=\) \(\displaystyle 66 \, 066\)
\(\displaystyle T_{1111}\) \(=\) \(\displaystyle 617 \, 716\)
\(\displaystyle T_{2662}\) \(=\) \(\displaystyle 3 \, 544 \, 453\)
\(\displaystyle T_{111 \, 111}\) \(=\) \(\displaystyle 6 \, 172 \, 882 \, 716\)
\(\displaystyle T_{246 \, 642}\) \(=\) \(\displaystyle 30 \, 416 \, 261 \, 403\)
\(\displaystyle T_{11 \, 111 \, 111}\) \(=\) \(\displaystyle 61 \, 728 \, 399 \, 382 \, 716\)
\(\displaystyle T_{363 \, 474 \, 363}\) \(=\) \(\displaystyle 66 \, 056 \, 806 \, 460 \, 865 \, 066\)
\(\displaystyle T_{2 \, 664 \, 444 \, 662}\) \(=\) \(\displaystyle 3 \, 549 \, 632 \, 679 \, 762 \, 369 \, 453\)
\(\displaystyle T_{26 \, 644 \, 444 \, 662}\) \(=\) \(\displaystyle 354 \, 963 \, 215 \, 686 \, 512 \, 369 \, 453\)
\(\displaystyle T_{246 \, 644 \, 446 \, 642}\) \(=\) \(\displaystyle 30 \, 416 \, 741 \, 529 \, 792 \, 514 \, 761 \, 403\)
\(\displaystyle T_{266 \, 444 \, 444 \, 662}\) \(=\) \(\displaystyle 35 \, 496 \, 321 \, 045 \, 754 \, 012 \, 369 \, 453\)
\(\displaystyle T_{2 \, 466 \, 444 \, 446 \, 642}\) \(=\) \(\displaystyle 3 \, 041 \, 674 \, 104 \, 186 \, 814 \, 014 \, 761 \, 403\)
\(\displaystyle T_{3 \, 654 \, 345 \, 456 \, 545 \, 434 \, 563}\) \(=\) \(\displaystyle 6 \, 677 \, 120 \, 357 \, 887 \, 130 \, 286 \, 820 \, 317 \, 887 \, 530 \, 217 \, 766\)

The sequence of the index elements is A008510 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The sequence of the triangular elements is A229236 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Historical Note

David Wells reports in Curious and Interesting Numbers, 2nd ed. that Charles Ashbacher reports on this sequence (in particular $363 \, 474 \, 363$) in Journal of Recreational Mathematics, Volume $24$, page $184$, but this has not been corroborated.


Sources