A350988 -id:A350988 - cp's OEIS frontend

A350987 Triangular numbers that are binary palindromes.

0, 1, 3, 15, 21, 45, 153, 231, 325, 561, 903, 2145, 4095, 8385, 14535, 33153, 58311, 63903, 101475, 131841, 240471, 399171, 525825, 932295, 976503, 1044735, 1308153, 1395285, 1532125, 2100225, 3727815, 8394753, 14680071, 17913105, 33566721, 54054003, 59650503

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since (2^k+1)*(2^k+2)/2 = A000217(A028401(k)) is a term for all k>1 (Trigg, 1974).

			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.
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.

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.

Intersection of A000217 and A006995.
The binary version of A003098.
A028401 \ {6} is a subsequence.
Cf. A350988, A350990 (base 3), A350991, A350992, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 2*10^4], PalindromeQ[IntegerDigits[#, 2]] &]

from itertools import count, islice
def agen():
    for i in count(0):
        t = i*(i+1)//2
        b = bin(t)[2:]
        if b == b[::-1]:
            yield t
print(list(islice(agen(), 37))) # Michael S. Branicky, Jan 28 2022

a(n) = A000217(A350988(n)).

A350989 Numbers k such that both k and the k-th triangular number are binary palindromes.

Original entry on oeis.org

0, 1, 5, 9, 17, 21, 33, 65, 129, 257, 341, 513, 693, 1025, 1365, 1397, 2049, 4097, 8193, 16385, 21845, 32769, 43605, 65537, 87125, 87381, 131073, 262145, 524289, 1048577, 1398101, 2097153, 2796885, 4194305, 5592405, 5594453, 8388609, 16777217, 33554433, 67108865

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since 2^k+1 is a term for all k>1.

			5 is a term since 5 = 101_2 is a binary palindromic number and A000217(5) = 5*(5+1)/2 = 15 = 1111_2 is a triangular number and also a binary palindromic number.

Amiram Eldar, Table of n, a(n) for n = 1..100

The binary version of A008510.
Intersection of A006995 and A350988.
A000051 \ {3} is a subsequence.
Cf. A229236, A350987.

Select[Range[0, 10^6], And @@ PalindromeQ /@ IntegerDigits[{#, #*(# + 1)/2}, 2] &]

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

from itertools import count, islice
def ispal(s): return s == s[::-1]
def ok(n): return ispal(bin(n)[2:]) and ispal(bin(n*(n+1)//2)[2:])
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 28 2022

cp's OEIS Frontend

A350987 Triangular numbers that are binary palindromes.

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