# Palindromic Triangular Numbers with Palindromic Index

## 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.