A350993 -id:A350993 - 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)).

A350990 Triangular numbers that are palindromes in base 3.

Original entry on oeis.org

0, 1, 10, 28, 91, 820, 7381, 65341, 66430, 597871, 1633528, 5380840, 48339028, 48427561, 139386556, 435848050, 1178284240, 3529890253, 3922632451, 32614707700, 35296517971, 35303692060, 101891588176, 292358957446, 295883935480, 317733228541, 859413596320, 2649105942220

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((3^k-1)/2) is a term for all k >= 0 (Trigg, 1971).

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 3: 10 = 101_3.
28 is a term since 28 = A000217(7) is a triangular number and also a palindromic number in base 3: 36 = 1001_3.

Amiram Eldar, Table of n, a(n) for n = 1..50
Charles W. Trigg, Problem 3413, Problem Department, School Science and Mathematics, Vol. 71, No. 9 (1971), p. 843; Solution to Problem 3413, by Bob Prielipp, Vol. 72, No. 4 (1972), p. 358.
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 A014190.
The ternary version of A003098.
Cf. A003462, A350987, A350991, A350992, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 3]] &]

A350991 Triangular numbers that are palindromes in base 5.

Original entry on oeis.org

0, 1, 3, 6, 36, 78, 378, 1953, 20706, 23436, 48828, 147696, 239778, 426426, 449826, 1220703, 2155926, 6011778, 14625936, 30517578, 74218836, 74316336, 149083278, 314290056, 351562386, 762939453, 7897542681, 9141750936, 10201418541, 19073486328, 35952613476, 38218245156

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((5^k-1)/2) is a term for all k >= 0 (Trigg, 1972).

			6 is a term since 6 = A000217(3) is a triangular number and also a palindromic number in base 5: 6 = 11_5.
36 is a term since 36 = A000217(8) is a triangular number and also a palindromic number in base 5: 36 = 121_5.

Amiram Eldar, Table of n, a(n) for n = 1..68
Charles W. Trigg, Problem 840, Problems and Solutions, Mathematics Magazine, Vol. 45, No. 4 (1972), p. 228; Triangular Palindromes, Solution to Problem 840 by E. P. Starke, ibid., Vol. 46, No. 3 (1973), p. 170.
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 A029952.
The quinary version of A003098.
Cf. A003098, A125831, A350987, A350990, A350992, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 5]] &]

A350992 Triangular numbers that are palindromes in base 7.

Original entry on oeis.org

0, 1, 3, 6, 78, 171, 300, 2850, 8256, 9453, 14706, 120786, 208335, 399171, 405450, 416328, 448878, 720600, 5877306, 6046503, 6835753, 9350650, 10122750, 18431556, 19130205, 22596003, 35309406, 499169406, 934394835, 969430528, 999335571, 1059265378, 1730160900

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((7^k-1)/2) is a term for all k >= 0 (Trigg, 1974).

			78 is a term since 78 = A000217(12) is a triangular number and also a palindromic number in base 7: 78 = 141_7.
171 is a term since 171 = A000217(18) is a triangular number and also a palindromic number in base 7: 171 = 333_7.

Amiram Eldar, Table of n, a(n) for n = 1..103
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.

Intersection of A000217 and A029954.
The septenary version of A003098.
Cf. A120741, A350987, A350990, A350991, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 7]] &]

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A350987 Triangular numbers that are binary palindromes.

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A350990 Triangular numbers that are palindromes in base 3.

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A350991 Triangular numbers that are palindromes in base 5.

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A350992 Triangular numbers that are palindromes in base 7.

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