A350987 -id:A350987 - cp's OEIS frontend

A350990 Triangular numbers that are palindromes in base 3.

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]] &]

A350993 Triangular numbers that are palindromes in base 9.

Original entry on oeis.org

0, 1, 3, 6, 10, 91, 136, 300, 528, 820, 4560, 7381, 11476, 20910, 42486, 66430, 552826, 581581, 597871, 1664400, 2001000, 3420420, 3444000, 5070520, 5380840, 48427561, 75995956, 132494781, 134553810, 137158203, 159213090, 290585778, 434520460, 435848050, 669615310

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((9^k-1)/2) is a term for all k >= 0 (Wishard, 1931).

Also, A000217((3 + 5*9^k)/2) is a term for all k>=0 (Trigg, 1984).

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 9: 10 = 11_9.
91 is a term since 91 = A000217(13) is a triangular number and also a palindromic number in base 9: 91 = 111_9.

Charles W. Trigg, Mathematical Quickies, McGraw Hill Book Co., 1967, Q112, p. 127.

Amiram Eldar, Table of n, a(n) for n = 1..123
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Charles W. Trigg, Problem 281, The College Mathematics Journal, Vol. 15, No. 4 (1984), p. 346; Palindromic Triangular Numbers in Base Nine, Solution to Problem 281, by Michael Vowe, ibid., Vol. 17, No. 2 (1986), pp. 188-189.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.
G. W. Wishard, Problem 3480, The American Mathematical Monthly, Vol. 38, No. 3 (1931), p. 170; Solution to Problem 3480, by Helen A. Merrill, ibid., Vol. 39, No. 3 (1932), p. 179.

Intersection of A000217 and A029955.
The nonary version of A003098.
Cf. A191681, A350987, A350990, A350991, A350992.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 9]] &]

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

Original entry on oeis.org

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

Amiram Eldar, Jan 28 2022

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

			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.

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.

The binary version of A008509.
A000051 \ {3} is a subsequence.
Cf. A000217, A003098, A006995, A350987.

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

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

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

A000217(a(n)) = A350987(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

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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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A350993 Triangular numbers that are palindromes in base 9.

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

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

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