A003098 -id:A003098 - cp's OEIS frontend

A008509 Positive integers k such that k-th triangular number is palindromic.

1, 2, 3, 10, 11, 18, 34, 36, 77, 109, 132, 173, 363, 1111, 1287, 1593, 1833, 2662, 3185, 3369, 3548, 8382, 11088, 18906, 50281, 57166, 102849, 111111, 167053, 179158, 246642, 337650, 342270, 365436, 417972, 1620621, 3240425, 3457634, 3707883

Offset: 1

Patrick De Geest

Charles W. Trigg, Palindromic Triangular Numbers, J. Rec. Math., 6 (1973), 146-147.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 93.

T. D. Noe, Table of n, a(n) for n = 1..147 (from P. De Geest)
P. De Geest, Palindromic Triangulars

Cf. A003098, A002113, A000217, A083571, A050721, A050722.

[k:k in [1..5000000]| Intseq(Binomial(k+1,2)) eq Reverse(Intseq(Binomial(k+1,2)))]; //  Marius A. Burtea, Jul 16 2019

palQ[n_]:= Reverse[x=IntegerDigits[n]]==x; t={}; Do[If[palQ[n*(n+1)/2],AppendTo[t,n]],{n,4*10^6}]; t (* Jayanta Basu, May 13 2013 *)
Position[Accumulate[Range[371*10^4]],?PalindromeQ]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 12 2020 *)

ispal(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1
is(n)=ispal(n*(n+1)/2) \\ Charles R Greathouse IV, May 15 2013

A035090 Non-palindromic squares which when written backwards remain square (and still have the same number of digits).

Original entry on oeis.org

144, 169, 441, 961, 1089, 9801, 10404, 10609, 12544, 12769, 14884, 40401, 44521, 48841, 90601, 96721, 1004004, 1006009, 1022121, 1024144, 1026169, 1042441, 1044484, 1062961, 1212201, 1214404, 1216609, 1236544, 1238769, 1256641

Offset: 1

Patrick De Geest, Nov 15 1998

Squares with trailing zeros not included.

Sequence is infinite, since it includes, e.g., 10^(2k) + 4*10^k + 4 for all k. - Robert Israel, Sep 20 2015

Robert Israel, Table of n, a(n) for n = 1..798
Patrick De Geest, Palindromic Squares in bases 2 to 17
Index entry for sequences related to reversing digits of squares

Reversing a polytopal number gives a polytopal number:
cube to cube: A035123, A035124, A035125, A002781;
square to square: A161902, A035090, A033294, A106323, A106324, A002779;
square to triangular: A181412, A066702;
tetrahedral to tetrahedral: A006030;
triangular to square: A066703, A179889;
triangular to triangular: A066528, A069673, A003098, A066569.
Cf. A319388.

rev:= proc(n) local L,i;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local t;
  if n mod 10 = 0 then return false fi;
  t:= rev(n);
t <> n and issqr(t)
end proc:
select(filter, [seq(n^2, n=1..10^5)]); # Robert Israel, Sep 20 2015

Select[Range[1200]^2,!PalindromeQ[#]&&IntegerLength[#]==IntegerLength[ IntegerReverse[ #]] && IntegerQ[Sqrt[IntegerReverse[#]]]&] (* Harvey P. Dale, Jul 19 2023 *)

a(n) = A035123(n)^2. - R. J. Mathar, Jan 25 2017

A069673 Nonpalindromic triangular numbers whose digit reversal is also a triangular number (possibly with fewer digits).

Original entry on oeis.org

10, 120, 153, 190, 300, 351, 630, 820, 17578, 87571, 156520, 180300, 185745, 547581, 557040, 678030, 1461195, 1851850, 5911641, 6056940, 12145056, 12517506, 16678200, 56440000, 60571521, 65054121, 157433640, 188267310, 304119453, 354911403, 1261250200

Offset: 1

Amarnath Murthy, Apr 06 2002

Giovanni Resta, Table of n, a(n) for n = 1..123 (terms < 2*10^24)

See A066528 for a different version.
Cf. A066703, A179889.
Cf. A003098, A061455, A066569.

More terms from Jason Earls, Jun 07 2002

a(27)-a(31) from Giovanni Resta, Jun 20 2015

A061455 Triangular numbers whose digit reversal is also a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 10, 55, 66, 120, 153, 171, 190, 300, 351, 595, 630, 666, 820, 3003, 5995, 8778, 15051, 17578, 66066, 87571, 156520, 180300, 185745, 547581, 557040, 617716, 678030, 828828, 1269621, 1461195, 1680861, 1851850, 3544453, 5073705, 5676765, 5911641

Offset: 1

Amarnath Murthy, May 03 2001

			153 is in the sequence because (1) it is a triangular number and (2) its reversal 351 is also a triangular number.

Jon E. Schoenfield, Table of n, a(n) for n = 1..162 (terms < 10^18)

Cf. A000217, A003098, A004086, A066528, A066569, A069673.

read("transforms");
isA000217 := proc(n) issqr(1+8*n) ;end proc:
isA061455 := proc(n) isA000217(n) and isA000217(digrev(n)) ; end proc:
for n from 0 to 60000 do T := A000217(n) ; if isA061455(T) then printf("%d,", T) ; end if; end do: # R. J. Mathar, Dec 13 2010

TriangularNumberQ[k_] := If[IntegerQ[1/2 (Sqrt[1 + 8 k] - 1)], True, False]; Select[Range[0, 5676765], TriangularNumberQ[#] && TriangularNumberQ[FromDigits[Reverse[IntegerDigits[#]]]] &] (* Ant King, Dec 13 2010 *)

isok(n) = ispolygonal(n, 3) && ispolygonal(fromdigits(Vecrev(digits(n))), 3); \\ Michel Marcus, Apr 14 2019

a(n)=A000217(k) and A004086(a(n))=A000217(j) for some k and j. - R. J. Mathar, Jun 02 2006

More terms from Erich Friedman, May 08 2001

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

A066528 Non-palindromic triangular numbers whose reverse is a triangular number with the same number of digits.

Original entry on oeis.org

153, 351, 17578, 87571, 185745, 547581, 1461195, 5911641, 12145056, 12517506, 60571521, 65054121, 304119453, 354911403, 1775275491, 1945725771, 10246462281, 17990863516, 18226464201, 35615002605, 50620051653, 61536809971, 1222080857271, 1664224065406

Offset: 1

Erich Friedman, Jan 08 2002

			153 and 351 are both triangular.

Giovanni Resta, Table of n, a(n) for n = 1..64 (terms < 2*10^24)

See A069673 for another version.
Cf. A066703, A179889.
Cf. A003098, A061455, A066569, A069673.

dtn[L_] := Fold[10#1+#2&, 0, L]; tritest[n_] := Module[{t}, t=Floor[N[Sqrt[2n]]]; 2n==t(t+1)]; A={}; For[i=1, i>0, i++, t=i(i+1)/2; If[tritest[tt=dtn[Reverse[IntegerDigits[t]]]]&&Mod[t, 10]>0&&t=!=tt, AppendTo[A, t]; Print[A]]]

a(22)-a(24) from Giovanni Resta, Jun 20 2015

A350987 Triangular numbers that are binary palindromes.

Original entry on oeis.org

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)).

A054263 Number of palindromic triangular numbers with n digits.

Original entry on oeis.org

3, 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

Offset: 1

Patrick De Geest, Apr 15 2000

			a(10) = 4: 1264114621, 1634004361, 5289009825, 6172882716.

P. De Geest, Palindromic Triangulars

Cf. A000217, A003098, A008509, A008510.

A066569 Triangular numbers whose reverse is also triangular.

Original entry on oeis.org

1, 3, 6, 55, 66, 153, 171, 351, 595, 666, 3003, 5995, 8778, 15051, 17578, 66066, 87571, 185745, 547581, 617716, 828828, 1269621, 1461195, 1680861, 3544453, 5073705, 5676765, 5911641, 6295926, 12145056, 12517506, 35133153, 60571521

Offset: 1

Erich Friedman, Jan 08 2002

Numbers ending in 0 are not included. - Harry J. Smith, Mar 06 2010

			153 and 351 are both triangular.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A003098, A061455, A066528, A069673.

dtn[L_] := Fold[10#1+#2&, 0, L] tritest[n_] := Module[{t}, t=Floor[N[Sqrt[2n]]]; 2n==t(t+1)] A={}; For[i=1, i>0, i++, t=i(i+1)/2; If[tritest[dtn[Reverse[IntegerDigits[t]]]]&&Mod[t, 10]>0, AppendTo[A, t]; Print[A]]]
Select[Accumulate[Range[12000]],Last[IntegerDigits[#]]!=0&&OddQ[Sqrt[1+ 8*FromDigits[Reverse[IntegerDigits[#]]]]]&] (* Harvey P. Dale, Jun 04 2015 *)

Rev(x)= { local(d, r=0); while (x>0, d=x%10; x\=10; r=r*10 + d); return(r) } { n=0; for (m=1, 10^10, t=m*(m + 1)/2; if (t%10 == 0, next); if (issquare(8*Rev(t) + 1), write("b066569.txt", n++, " ", t); if (n==100, return)) ) } \\ Harry J. Smith, Mar 08 2010

Offset changed from 0 to 1 by Harry J. Smith, Mar 06 2010

A226742 Triangular numbers obtained as the concatenation of 2*k and k.

Original entry on oeis.org

21, 105, 2211, 9045, 222111, 306153, 742371, 890445, 1050525, 22221111, 88904445, 107905395, 173808690, 2222211111, 8889044445, 12141260706, 15754278771, 222222111111, 888890444445, 22222221111111, 36734701836735, 65306123265306

Offset: 1

Antonio Roldán, Jun 18 2013

Includes (2*10^k+1)*(10^k-1)/9 and (2*10^k+1)*(4*10^k+5)/9 for k >= 1. - Robert Israel, Feb 06 2025

			If k=111, 2k=222, 2k//k = 222111 = 666*667/2, a triangular number.

Robert Israel, Table of n, a(n) for n = 1..2675

Cf. A003098, A068899, A226772, A226788, A226789, A380792.

g:= proc(d) local a, b, n, Res, x, y;
      Res:= NULL:
      for a in numtheory:-divisors(2*(2*10^d+1)) do
        b:= 2*(2*10^d+1)/a;
        if igcd(a, b)>1 then next fi;
        n:= chrem([0, -1], [a, b]);
        x:= n*(n+1)/2;
        y:= x/(2*10^d+1);
        if y < 10^(d-1) or y >= 10^d  then next fi;
        Res:= Res, (2*10^d+1)*y
      od;
      op(sort([Res]))
end proc:
map(g, [$1..10]); # Robert Israel, Feb 06 2025

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[2*n], IntegerDigits[n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)

concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^5,a=concatint(2*n,n);if(istriang(a),print(a)))}

A226772 Triangular numbers obtained as the concatenation of n and 2n.

Original entry on oeis.org

36, 1326, 2346, 3570, 125250, 223446, 12502500, 22234446, 1250025000, 2066441328, 2222344446, 2383847676, 3673573470, 125000250000, 222223444446, 5794481158896, 12500002500000, 12857132571426, 22222234444446, 49293309858660, 804878916097578, 933618918672378, 971908519438170

Offset: 1

Antonio Roldán, Jun 18 2013

Includes 125*10^(2*k+1)+25*10^k and (10^k+2)*(1+(10^k-1)*2/9) for k >= 1. - Robert Israel, Nov 09 2020

			If n=23, 2n=46, n//2n = 2346 = 68*69/2, a triangular number.

Robert Israel, Table of n, a(n) for n = 1..1509

Cf. A003098, A068899, A226742, A226788, A226789.

F:= proc(d) local D,R,M,m,w,x,x1,x2;
   R:= NULL;
   M:= 10^d/2+1;
   D:= numtheory:-divisors(M);
   for m in D do if igcd(m,M/m)=1 then
     for w in [chrem([-1,1],[8*m,M/m]), chrem([1,-1],[8*m,M/m])] do
     x:= (w^2-1)/8;
     x1:= x mod 10^d;
     x2:= floor(x/10^d);
     if x1 = 2*x2 and x1 >= 10^(d-1) then R:= R, x fi
   od fi od;
   op(sort([R]))
end proc:
36, seq(F(d),d=2..10); # Robert Israel, Nov 09 2020

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n], IntegerDigits[2*n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)

concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^5,a=concatint(n,2*n);if(istriang(a),print(a)))}

cp's OEIS Frontend

A008509 Positive integers k such that k-th triangular number is palindromic.

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A035090 Non-palindromic squares which when written backwards remain square (and still have the same number of digits).

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A069673 Nonpalindromic triangular numbers whose digit reversal is also a triangular number (possibly with fewer digits).

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A061455 Triangular numbers whose digit reversal is also a triangular number.

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A066528 Non-palindromic triangular numbers whose reverse is a triangular number with the same number of digits.

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A350987 Triangular numbers that are binary palindromes.

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A054263 Number of palindromic triangular numbers with n digits.

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A066569 Triangular numbers whose reverse is also triangular.

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