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

1,3

Comments

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

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.

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

The binary version of A008509.

A000051 \ {3} is a subsequence.

Cf. A000217, A003098, A006995, A350987.

Sequence in context: A301791 A051677 A122965 * A226810 A054463 A295741

Adjacent sequences: A350985 A350986 A350987 * A350989 A350990 A350991

Keyword

nonn,base

Author

Amiram Eldar, Jan 28 2022

Status

approved