A003098 Palindromic triangular numbers. (Formerly M2605)

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

1,3

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.

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 *)

Prog

(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

Crossrefs

Cf. A008509.

Intersection of A000217 and A002113.

Sequence in context: A355637 A066569 A051641 * A045914 A370520 A303351

Adjacent sequences: A003095 A003096 A003097 * A003099 A003100 A003101

Keyword

nonn,base,nice,changed

Author

N. J. A. Sloane

Status

approved