A350993 Triangular numbers that are palindromes in base 9.

0, 1, 3, 6, 10, 91, 136, 300, 528, 820, 4560, 7381, 11476, 20910, 42486, 66430, 552826, 581581, 597871, 1664400, 2001000, 3420420, 3444000, 5070520, 5380840, 48427561, 75995956, 132494781, 134553810, 137158203, 159213090, 290585778, 434520460, 435848050, 669615310

Offset: 1

Author

Amiram Eldar, Jan 28 2022

Keywords

• nonn
• base

Comments

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

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

Examples

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 9: 10 = 11_9.
91 is a term since 91 = A000217(13) is a triangular number and also a palindromic number in base 9: 91 = 111_9.
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..123
• Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
• Charles W. Trigg, Problem 281, The College Mathematics Journal, Vol. 15, No. 4 (1984), p. 346; Palindromic Triangular Numbers in Base Nine, Solution to Problem 281, by Michael Vowe, ibid., Vol. 17, No. 2 (1986), pp. 188-189.
• Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.
• G. W. Wishard, Problem 3480, The American Mathematical Monthly, Vol. 38, No. 3 (1931), p. 170; Solution to Problem 3480, by Helen A. Merrill, ibid., Vol. 39, No. 3 (1932), p. 179.

Crossrefs

Intersection of A000217 and A029955.

The nonary version of A003098.

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

Programs

Mathematica

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