cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Comments

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

Links

Crossrefs

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.

Programs

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

Formula

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

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

Views

Author

Amarnath Murthy, May 03 2001

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    isok(n) = ispolygonal(n, 3) && ispolygonal(fromdigits(Vecrev(digits(n))), 3); \\ Michel Marcus, Apr 14 2019

Formula

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

Extensions

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

Views

Author

Erich Friedman, Jan 08 2002

Keywords

Examples

			153 and 351 are both triangular.

Links

Crossrefs

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

Programs

  • Mathematica
    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]]]

Extensions

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

A066703 Triangular numbers whose reverse is a square with the same number of digits.

Original entry on oeis.org

0, 1, 9446031, 1270004401, 14214075921, 1809702709101, 4614899724711, 6766532724546, 52657436563056, 98855178542676, 520454948099628321, 467756399278821844071, 40441102744430519189191
Offset: 1

Views

Author

Erich Friedman, Jan 14 2002

Keywords

Comments

The sequence of corresponding squares is A066702. - Robert G. Wilson v, Jan 31 2011
a(14) > 2*10^24. - Giovanni Resta, Jun 20 2015

Examples

			9446031 is triangular and 1306449 is a square.

Crossrefs

See A179889 for another version. Cf. A066702, A069673.

Programs

  • Mathematica
    lst = {0}; For[i=1, i<10^6, i++, t=i(i+1)/2; r=FromDigits@ Reverse@ IntegerDigits@ t; If[ Mod[t, 10] > 0 && IntegerQ@ Sqrt@ r, AppendTo[lst, t]; Print@ lst]]

Extensions

More terms from Jason Earls and the author, Jan 15 2002
Offset and definition modified at the suggestion of Harvey P. Dale, Jan 30 2011
a(11) from Donovan Johnson, Jan 31 2011
a(12)-a(13) from Giovanni Resta, Jun 20 2015

A179889 Triangular numbers whose reverse is a square (possibly with fewer digits).

Original entry on oeis.org

1, 10, 630, 52650, 165600, 986310, 9446031, 9485190, 10693000, 1270004401, 14214075921, 140884670790, 1809702709101, 4614899724711, 6766532724546, 9802814901400, 10210140486640, 14287075542460, 52657436563056, 98855178542676
Offset: 1

Views

Author

Harvey P. Dale, Jan 30 2011

Keywords

Examples

			9446031 is triangular and 1306449 is a square.

Links

Crossrefs

A variant of A066703. Cf. A069673, A181412, A066528.

Programs

  • Mathematica
    trnos=Accumulate[Range[14070000]];
sqnoQ[n_]:=IntegerQ[Sqrt[FromDigits[Reverse[IntegerDigits[n]]]]]
Select[trnos,sqnoQ]  (* Harvey P. Dale, Jan 31 2011 *)

Extensions

More terms from Harvey P. Dale, Jan 31 2011

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

Views

Author

Erich Friedman, Jan 08 2002

Keywords

Comments

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

Examples

			153 and 351 are both triangular.

Links

Crossrefs

Cf. A003098, A061455, A066528, A069673.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

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

A181412 Squares whose reverse is a triangular number; trailing zeros are permitted.

Original entry on oeis.org

1, 100, 10000, 1000000, 1306449, 100000000, 130644900, 1044000721, 10000000000, 12957041241, 13064490000, 104400072100, 1000000000000, 1019072079081, 1174279984164, 1295704124100, 1306449000000, 6454272356676, 10440007210000
Offset: 1

Views

Author

Harvey P. Dale, Jan 30 2011

Keywords

Comments

Suggested by T. D. Noe.

Examples

			1306449 is 1143 squared, and its reverse, 9446031, is a triangular number.

Links

Crossrefs

Cf. A066702, A001110, A066703, A179889, A066528, A069673.

Programs

  • Mathematica
    trnos = Accumulate[Range[300000]]; Select[Range[210000]^2, MemberQ[trnos, FromDigits[Reverse[IntegerDigits[#]]]] &]

Extensions

a(12)-a(19) from Donovan Johnson, Feb 12 2011
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.