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

%I A350988 #17 Jan 05 2025 19:51:42
%S A350988 0,1,2,5,6,9,17,21,25,33,42,65,90,129,170,257,341,357,450,513,693,893,
%T A350988 1025,1365,1397,1445,1617,1670,1750,2049,2730,4097,5418,5985,8193,
%U A350988 10397,10922,16385,17313,21717,21845,31749,32769,40637,43605,51537,63482,65537,76217
%N A350988 Numbers k such that the k-th triangular number is a binary palindrome.
%C A350988 This sequence is infinite since 2^k+1 is a term for all k>1 (Trigg, 1974).
%H A350988 Amiram Eldar, <a href="/A350988/b350988.txt">Table of n, a(n) for n = 1..128</a>
%H A350988 Charles W. Trigg, <a href="https://web.archive.org/web/2024*/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 A350988 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.
%F A350988 A000217(a(n)) = A350987(n).
%e A350988 2 is a term since A000217(2) = 2*(2+1)/2 = 3 = 11_2 is a triangular number and also a binary palindromic number.
%e A350988 5 is a term since A000217(5) = 5*(5+1)/2 = 15 = 1111_2 is a triangular number and also a binary palindromic number.
%t A350988 Select[Range[0, 10^5], PalindromeQ[IntegerDigits[#*(# + 1)/2, 2]] &]
%o A350988 (PARI) isok(k) = my(b=binary(k*(k+1)/2)); b == Vecrev(b); \\ _Michel Marcus_, Jan 28 2022
%o A350988 (Python)
%o A350988 def ok(n): b = bin(n*(n+1)//2)[2:]; return b == b[::-1]
%o A350988 print([k for k in range(80000) if ok(k)]) # _Michael S. Branicky_, Jan 28 2022
%Y A350988 The binary version of A008509.
%Y A350988 A000051 \ {3} is a subsequence.
%Y A350988 Cf. A000217, A003098, A006995, A350987.
%K A350988 nonn,base
%O A350988 1,3
%A A350988 _Amiram Eldar_, Jan 28 2022