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A350987 Triangular numbers that are binary palindromes.

%I A350987 #19 Jan 05 2025 19:51:42
%S A350987 0,1,3,15,21,45,153,231,325,561,903,2145,4095,8385,14535,33153,58311,
%T A350987 63903,101475,131841,240471,399171,525825,932295,976503,1044735,
%U A350987 1308153,1395285,1532125,2100225,3727815,8394753,14680071,17913105,33566721,54054003,59650503
%N A350987 Triangular numbers that are binary palindromes.
%C A350987 This sequence is infinite since (2^k+1)*(2^k+2)/2 = A000217(A028401(k)) is a term for all k>1 (Trigg, 1974).
%H A350987 Amiram Eldar, <a href="/A350987/b350987.txt">Table of n, a(n) for n = 1..128</a>
%H A350987 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 A350987 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 A350987 a(n) = A000217(A350988(n)).
%e A350987 3 is a term since 3 = A000217(2) = 2*(2+1)/2 is a triangular number, and 3 = 11_2 is also a binary palindromic number.
%e A350987 15 is a term since 15 = A000217(5) = 5*(5+1)/2 is a triangular number, and 15 = 1111_2 is also a binary palindromic number.
%t A350987 t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 2*10^4], PalindromeQ[IntegerDigits[#, 2]] &]
%o A350987 (Python)
%o A350987 from itertools import count, islice
%o A350987 def agen():
%o A350987     for i in count(0):
%o A350987         t = i*(i+1)//2
%o A350987         b = bin(t)[2:]
%o A350987         if b == b[::-1]:
%o A350987             yield t
%o A350987 print(list(islice(agen(), 37))) # _Michael S. Branicky_, Jan 28 2022
%Y A350987 Intersection of A000217 and A006995.
%Y A350987 The binary version of A003098.
%Y A350987 A028401 \ {6} is a subsequence.
%Y A350987 Cf. A350988, A350990 (base 3), A350991, A350992, A350993.
%K A350987 nonn,base
%O A350987 1,3
%A A350987 _Amiram Eldar_, Jan 28 2022