A003098 -id:A003098 - cp's OEIS frontend

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

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).

A229236 Palindromic triangular number with palindromic index.

Original entry on oeis.org

1, 3, 6, 66, 3003, 66066, 617716, 3544453, 6172882716, 30416261403, 61728399382716, 66056806460865066, 3549632679762369453, 354963215686512369453, 30416741529792514761403, 35496321045754012369453, 3041674104186814014761403

Offset: 1

Shyam Sunder Gupta, Sep 17 2013

			a(5)=3003 because 3003 is 77th triangular number and 77 is palindrome.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..18

Cf. A003098, A000217, A008509, A008510.

Select[Table[{n,(n(n+1))/2},{n,10^8}],AllTrue[#,PalindromeQ]&][[;;,2]] (* Harvey P. Dale, Jun 04 2023 *)

Previous Mathematica program replaced by Harvey P. Dale, Jun 04 2023

A307348 Numbers k such that there are no palindromic triangular numbers of length k.

Original entry on oeis.org

12, 30, 40

Offset: 1

Robert Price, May 01 2019

Patrick De Geest, Palindromic Squares in bases 2 to 17
Eric Weisstein's World of Mathematics, Palindromic Number

Cf. A000217, A003098, A008509, A034822.

A003098 = Select[PolygonalNumber[3, Range[0, 10^6]], PalindromeQ];
(* Set Range to level of desired running time. *)
nopal[n_] := Length[Select[A003098, IntegerLength[#] == n || (n == 1 && # == 0) &]]; Select[Range[12], nopal[#] == 0 &]
(* Set Range to encompass length of last value in A003098. *)

A307367 Number of palindromic triangular numbers with exactly n digits.

Original entry on oeis.org

4, 2, 3, 3, 2, 2, 6, 2, 1, 4, 7, 0, 4, 4, 12, 5, 6, 2, 3, 2, 6, 3, 6, 2, 2, 4, 3, 2, 5, 0, 3, 2, 1, 4, 3, 1, 10, 1, 4, 0, 3, 2, 2, 1, 1

Offset: 1

Robert Price, May 01 2019

Number of terms in A003098 with exactly n digits.

Differs from A054263 only at a(1), assuming 0 has 1 digits. - R. J. Mathar, May 06 2019

G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy] See page 95.

Cf. A000217, A003098, A008509, A034822, A263618, A307348.

A003098 = Select[PolygonalNumber[3, Range[0, 10^6]], PalindromeQ];
  (* Set Range to level of desired running time. \ *)
Table[Length[ Select[A003098, IntegerLength[#] == n || (n == 1 && # == 0) &]], {n, 12}]
  (* Set Range to encompass length of last value in A003098. *)

A088278 Smallest palindromic triangular numbers beginning with palindromes whose first digit is 1, 3, 5, 6, or 8.

Original entry on oeis.org

1, 3, 55, 6, 8778, 114401848104411, 0, 55, 66, 0, 10129457886113466431168875492101, 11121736463712111

Offset: 1

Amarnath Murthy, Sep 29 2003

The possible values of the ones digit of a triangular number are 0,1,3,5,6 and 8. Similarly, one can list the two-digit numbers k such that a triangular number of the form 100r + k can exist, and so on for the first three digits, etc. For palindromes P beginning with numbers other than these (e.g., for 33 and 88, which are two-digit palindromes P that start with 1, 3, 5, 6, or 8 but are not in A187127), the corresponding term is 0.

			From _Jon E. Schoenfield_, Mar 03 2018: (Start)
        Palindrome P       a(n) = smallest palindromic
       starting with    triangular number starting with P
   n  1, 3, 5, 6, or 8   (or 0 if no such number exists)
  ==  ================  =================================
   1          1                                         1
   2          3                                         3
   3          5                                        55
   4          6                                         6
   5          8                                      8778
   6         11                           114401848104411
   7         33                                         0
   8         55                                        55
   9         66                                        66
  10         88                                         0
  11        101          10129457886113466431168875492101
  12        111                         11121736463712111
  13        121                                         ?
  14        131                             1313207023131
  15        141                                         ?
  16        151                      15199896744769899151
  17        161                                         ?
  18        171                                       171
  19        181                                         ?
  20        191                                         ?
  21        303                                         0
(End)

Cf. A003098 (palindromic triangular numbers), A187127 (numbers that are the residue mod 100 of a triangular number). - Jon E. Schoenfield, Mar 03 2018

Name and Comments edited, offset changed to 1, and a(11) and a(12) corrected (a(11) taken from b-file at A003098) by Jon E. Schoenfield, Mar 03 2018

A101034 Numbers n such that A002113(n) is a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 14, 15, 26, 68, 75, 129, 158, 186, 249, 759, 1616, 1827, 2268, 2679, 4543, 6072, 6675, 7294, 13512, 16146, 27871, 112640, 116339, 152889, 161727, 239533, 260487, 404161, 670038, 685744, 767718, 973504, 2313206, 6250177, 6977617

Offset: 1

Klaus Brockhaus, Nov 27 2004

Indices of triangular numbers in the sequence of palindromes.

			A002113(26) = 171 is a triangular number, so 26 is a term.

Cf. A000217, A002113, A003098, A008509.

var c,n,m:integer; end; begin c:=0; for n:=0 to 100000000 do if n = intreverse(n) then m:=floor(sqrt(2*n)); if m*(m+1) div 2 = n then write(c,","); end; inc(c); end; end; end;

Join[{0},Flatten[Position[Select[Range[10^7],PalindromeQ],?(OddQ[Sqrt[ 8#+1]]&)]]] (* The program generates the first 22 terms of the sequence. *) (* _Harvey P. Dale, Jun 17 2022 *)

{c=0;for(n=0,10000000,k=n;r=0;while(k>0,d=divrem(k,10);k=d[1];r=10*r+d[2]); if(n==r,m=sqrtint(2*n);if(m*(m+1)/2==n,print1(c,","));c++))}

a(24) to a(38) from Klaus Brockhaus, Oct 05 2005

A307850 Number of palindromic triangular numbers of length n whose index is also palindromic.

Original entry on oeis.org

4, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Robert Price, May 01 2019

			There is only one palindromic triangular number of length 2 whose index is also palindromic. 11->66. Thus, a(2)=1.

Patrick De Geest, Palindromic Squares in bases 2 to 17
Eric Weisstein's World of Mathematics, Palindromic Number

Cf. A000217, A003098, A008509, A307348, A307367.

A003098 = Select[PolygonalNumber[3, Range[0, 10^6]], PalindromeQ]
  (* Set Range to level of desired running time. *)
A008509 = Select[Range[0, 10^6], PalindromeQ[PolygonalNumber[3, #]] &]
  (* Set Range to level of desired running time. *)
Table[Length[ Select[A008509[[Table[ Select[Range[35], IntegerLength[A003098[[#]]] == n || (n == 1 && A003098[[#]] == 0) &], {35}][[n]]]], PalindromeQ[#] &]], {n, 11}]
  (* Set the first two Ranges to encompass the length of A003098 and  the last Range to encompass the length of the last value in A003098. *)

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

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A229236 Palindromic triangular number with palindromic index.

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A307348 Numbers k such that there are no palindromic triangular numbers of length k.

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A307367 Number of palindromic triangular numbers with exactly n digits.

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A088278 Smallest palindromic triangular numbers beginning with palindromes whose first digit is 1, 3, 5, 6, or 8.

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A101034 Numbers n such that A002113(n) is a triangular number.

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ARIBAS

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A307850 Number of palindromic triangular numbers of length n whose index is also palindromic.

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Mathematica