A003098 Palindromic triangular numbers.

0, 1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15051, 66066, 617716, 828828, 1269621, 1680861, 3544453, 5073705, 5676765, 6295926, 35133153, 61477416, 178727871, 1264114621, 1634004361, 5289009825, 6172882716, 13953435931

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice

Comments

The only known terms with an even number 2*m of digits that are the concatenation of two palindromes with m digits are 55, 66 and 828828 (see David Wells entry 828828). - Bernard Schott, Apr 29 2022

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Charles W. Trigg, Palindromic Triangular Numbers, J. Rec. Math., 6 (1973), 146-147.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 73 and p. 178, entry 828828 (Rev. ed. 1997)

Links

• T. D. Noe, Table of n, a(n) for n = 1..148 (from Patrick De Geest)
• Patrick De Geest, Palindromic Triangulars
• Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Crossrefs

Cf. A008509.

Intersection of A000217 and A002113.

Programs

Mathematica

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[ Accumulate[ Range[200000]],palQ]  (* Harvey P. Dale, Mar 23 2011 *)
Select[Accumulate[Range[0,170000]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

PARI

list(lim)=my(v=List(),d); for(n=0,(sqrt(8*lim+1)-1)/2, d=digits(n*(n+1)/2); if(d==Vecrev(d), listput(v,n*(n+1)/2))); Vec(v) \\ Charles R Greathouse IV, Jun 23 2017

Python

A003098_list = [m for m in (n*(n+1)//2 for n in range(10**5)) if str(m) == str(m)[::-1]] # Chai Wah Wu, Sep 03 2021