A350993 - OEIS

%I #16 Jan 18 2024 09:58:14

%S 0,1,3,6,10,91,136,300,528,820,4560,7381,11476,20910,42486,66430,

%T 552826,581581,597871,1664400,2001000,3420420,3444000,5070520,5380840,

%U 48427561,75995956,132494781,134553810,137158203,159213090,290585778,434520460,435848050,669615310

%N Triangular numbers that are palindromes in base 9.

%C This sequence is infinite since A000217((9^k-1)/2) is a term for all k >= 0 (Wishard, 1931).

%C Also, A000217((3 + 5*9^k)/2) is a term for all k>=0 (Trigg, 1984).

%D Charles W. Trigg, Mathematical Quickies, McGraw Hill Book Co., 1967, Q112, p. 127.

%H Amiram Eldar, <a href="/A350993/b350993.txt">Table of n, a(n) for n = 1..123</a>

%H Charles W. Trigg, <a href="https://www.fq.math.ca/Scanned/12-2/trigg.pdf">Infinite sequences of palindromic triangular numbers</a>, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.

%H Charles W. Trigg, <a href="https://www.jstor.org/stable/2686405">Probelm 281</a>, The College Mathematics Journal, Vol. 15, No. 4 (1984), p. 346; <a href="http://www.jstor.org/stable/2686843">Palindromic Triangular Numbers in Base Nine</a>, Solution to Problem 281, by Michael Vowe, ibid., Vol. 17, No. 2 (1986), pp. 188-189.

%H Maciej Ulas, <a href="https://arxiv.org/abs/0811.2477">On certain diophantine equations related to triangular and tetrahedral numbers</a>, arXiv:0811.2477 [math.NT], 2008.

%H G. W. Wishard, <a href="https://www.jstor.org/stable/2300979">Problem 3480</a>, The American Mathematical Monthly, Vol. 38, No. 3 (1931), p. 170; <a href="https://www.jstor.org/stable/2302299">Solution to Problem 3480</a>, by Helen A. Merrill, ibid., Vol. 39, No. 3 (1932), p. 179.

%e 10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 9: 10 = 11_9.

%e 91 is a term since 91 = A000217(13) is a triangular number and also a palindromic number in base 9: 91 = 111_9.

%t t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 9]] &]

%Y Intersection of A000217 and A029955.

%Y The nonary version of A003098.

%Y Cf. A191681, A350987, A350990, A350991, A350992.

%K nonn,base

%O 1,3

%A _Amiram Eldar_, Jan 28 2022