A350988 Numbers k such that the k-th triangular number is a binary palindrome.

0, 1, 2, 5, 6, 9, 17, 21, 25, 33, 42, 65, 90, 129, 170, 257, 341, 357, 450, 513, 693, 893, 1025, 1365, 1397, 1445, 1617, 1670, 1750, 2049, 2730, 4097, 5418, 5985, 8193, 10397, 10922, 16385, 17313, 21717, 21845, 31749, 32769, 40637, 43605, 51537, 63482, 65537, 76217

Offset: 1

Author

Amiram Eldar, Jan 28 2022

Keywords

• nonn
• base

Comments

This sequence is infinite since 2^k+1 is a term for all k>1 (Trigg, 1974).

Examples

			2 is a term since A000217(2) = 2*(2+1)/2 = 3 = 11_2 is a triangular number and also a binary palindromic number.
5 is a term since A000217(5) = 5*(5+1)/2 = 15 = 1111_2 is a triangular number and also a binary palindromic number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..128
• Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
• Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

Crossrefs

The binary version of A008509.

A000051 \ {3} is a subsequence.

Cf. A000217, A003098, A006995, A350987.

Programs

Mathematica

Select[Range[0, 10^5], PalindromeQ[IntegerDigits[#*(# + 1)/2, 2]] &]

PARI

isok(k) = my(b=binary(k*(k+1)/2)); b == Vecrev(b); \\ Michel Marcus, Jan 28 2022

Python

def ok(n): b = bin(n*(n+1)//2)[2:]; return b == b[::-1]
print([k for k in range(80000) if ok(k)]) # Michael S. Branicky, Jan 28 2022

Formula

A000217(a(n)) = A350987(n).