A002779 -id:A002779 - cp's OEIS frontend

A002778 Numbers whose square is a palindrome.

0, 1, 2, 3, 11, 22, 26, 101, 111, 121, 202, 212, 264, 307, 836, 1001, 1111, 2002, 2285, 2636, 10001, 10101, 10201, 11011, 11111, 11211, 20002, 20102, 22865, 24846, 30693, 100001, 101101, 110011, 111111, 200002, 798644, 1000001, 1001001

Offset: 1

N. J. A. Sloane

A002779(n) = a(n)^2; A136522(A000290(a(n))) = 1. - Reinhard Zumkeller, Oct 11 2011

See A016113 for the subset of numbers whose palindromic squares have an even number of digits. - M. F. Hasler, Jun 08 2014

			26^2 = 676, which is a palindrome, so 26 is in the sequence.
27^2 = 729, which is not a palindrome, so 27 is not in the sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Table of n, a(n) for n = 1..8729 (communicated by Max Alekseyev)
Patrick De Geest, Table of square palindromes
Patrick De Geest, Palindromic Squares in bases 2 to 17
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]
William Rex Marshall, Palindromic Squares
Gustavus J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
Gustavus J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Palindromic Number.
Feng Yuan, Palindromic Square Numbers

Cf. A002779, A002113, A016113, A136522, A000290.
See A003166 for binary analog.
For analogs in bases 2,3,4,5,etc. see A003166 onwards, A029984 onwards, and A263607 onwards.

a002778 n = a002778_list !! (n-1)
a002778_list = filter ((== 1) . a136522 . (^ 2)) [0..]
-- Reinhard Zumkeller, Oct 11 2011

[n: n in [0..2*10^6] | Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Apr 07 2015

palsquareQ[n_] := (n2 = IntegerDigits[n^2]; n2 == Reverse[n2]); A002778 = {}; Do[ If[palsquareQ[n], Print[n]; AppendTo[A002778, n]], {n, 0, 2 * 10^6}]; A002778 (* Jean-François Alcover, Dec 01 2011 *)
Sqrt[#]&/@Select[Range[0, 12 * 10^5]^2, # == IntegerReverse[#] &] (* The program uses the IntegerReverse function from Mathematica version 10. - Harvey P. Dale, Mar 04 2016 *)
Select[Range[0, 1001001], PalindromeQ[#^2] &] (* Michael De Vlieger, Dec 06 2017 *)

is_A002778(n)=is_A002113(n^2) \\ M. F. Hasler, Jun 08 2014

from itertools import count, islice
def A002778_gen(): # generator of terms
    return filter(lambda k: (s:=str(k**2))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A002778_list = list(islice(A002778_gen(),20)) # Chai Wah Wu, Jun 23 2022

More terms from Patrick De Geest

A033934 a(n) = (10^n + 1)^2.

Original entry on oeis.org

4, 121, 10201, 1002001, 100020001, 10000200001, 1000002000001, 100000020000001, 10000000200000001, 1000000002000000001, 100000000020000000001, 10000000000200000000001, 1000000000002000000000001, 100000000000020000000000001, 10000000000000200000000000001

Offset: 0

Jeff Burch

The members of this sequence are both perfect squares and palindromes. Therefore A002779 is an infinite sequence. - Ant King, Jun 26 2011

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002779 (palindromic squares), A000290 (squares), A002113 (palindromes).

Cf. A011557, A062397, A066138.

(10^Range[0,20]+1)^2 (* or *) LinearRecurrence[{111,-1110,1000},{4,121,10201},20] (* Harvey P. Dale, Feb 16 2016 *)

my(x='x+O('x^15)); Vec((1210*x^2-323*x+4)/(-1000*x^3+1110*x^2-111*x+1)) \\ Elmo R. Oliveira, Jul 04 2025

a(n) = A062397(n)^2 = A066138(n) + A011557(n).
From Elmo R. Oliveira, Jul 04 2025: (Start)
G.f.: (4 - 323*x + 1210*x^2)/((1-x)*(1-10*x)*(1-100*x)).
E.g.f.: exp(x)*(1 + 2*exp(9*x) + exp(99*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). (End)

Better description from Henry Bottomley, Dec 07 2001

More terms from Harvey P. Dale, Feb 16 2016

A034822 Numbers k such that there are no palindromic squares of length k.

Original entry on oeis.org

2, 4, 8, 10, 14, 18, 20, 24, 30, 38, 40

Offset: 1

Patrick De Geest, Oct 15 1998

All terms are even since (10^k+1)^2 is a palindrome of length 2*k+1. a(12) >= 46 if it exists (see A263618). - Chai Wah Wu, Jun 14 2024

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

Cf. A002778, A002779, A034307, A263618.

A034822[n_] := Select[Range[Ceiling[Sqrt[10^(n - 1)]], Floor[Sqrt[10^n]]], #^2 == IntegerReverse[#^2] &];
Select[Range[12], Length[A034822[#]] == 0 &] (* Robert Price, Apr 23 2019 *)

from sympy import integer_nthroot as iroot
def ispal(n): s = str(n); return s == s[::-1]
def ok(n):
  for r in range(iroot(10**(n-1), 2)[0] + 1, iroot(10**n, 2)[0]):
    if ispal(r*r): return False
  return True
print([m for m in range(1, 16) if ok(m)]) # Michael S. Branicky, Feb 04 2021

Two more terms from Patrick De Geest, Apr 01 2002

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

A029734 Palindromic squares in base 16.

Original entry on oeis.org

0, 1, 4, 9, 289, 1156, 66049, 74529, 83521, 93025, 101761, 264196, 280900, 298116, 597529, 1896129, 16785409, 19088161, 21538881, 67141636, 68128516, 71673156, 76282756, 81830116, 2665553641, 4295098369, 4328718849

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..377 (terms 1..75 from Vincenzo Librandi)
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb16Q[n_]:=Module[{idn16=IntegerDigits[n,16]}, idn16==Reverse[idn16]]; Select[Range[0, 200000]^2, pb16Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029738 Palindromic squares in base 12.

Original entry on oeis.org

0, 1, 4, 9, 169, 676, 21025, 24649, 28561, 32041, 32761, 84100, 85264, 91204, 373321, 2989441, 3045025, 3179089, 3553225, 4165681, 11957764, 13060996, 14409616, 430023169, 436016161, 442050625, 448126561, 505215529

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..58
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb12Q[n_]:=Module[{idn12=IntegerDigits[n, 12]}, idn12==Reverse[idn12]]; Select[Range[0, 20000]^2, pb12Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029806 n in base 8 is a palindromic square.

Original entry on oeis.org

0, 1, 4, 9, 36, 81, 121, 729, 4225, 5329, 6241, 6561, 6889, 38025, 47961, 56169, 133956, 263169, 294849, 342225, 485809, 1196836, 2368521, 3080025, 3515625, 8468100, 16785409, 17313921, 17850625, 20043529, 21316689, 21911761

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb8Q[n_]:=Module[{idn8=IntegerDigits[n, 8]}, idn8==Reverse[idn8]]; Select[Range[0, 20000]^2, pb8Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029991 Squares which are palindromes in base 6.

Original entry on oeis.org

0, 1, 4, 49, 1369, 1849, 5776, 8281, 47089, 67081, 1132096, 1682209, 1776889, 2307361, 2418025, 7311616, 11336689, 60481729, 64464841, 82428241, 389944009, 2176875649, 2197078129, 2302944121, 2323722025, 2963604721

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..92
Patrick De Geest, Palindromic Squares in bases 2 to 17

Squares which are palindromes in base b: A029983 (b=2), A029985 (b=3), A029987 (b=4), A029989 (b=5), this sequence (b=6), A029993 (b=7), A029806 (b=8), A029995 (b=9), A002779 (b=10), A029997 (b=11), A029738 (b=12), A029999 (b=13), A030074 (b=14), A030075 (b=15), A029734 (b=16).

Cf. A029990.

a(n) = A029990(n)^2. - Seiichi Manyama, Oct 16 2021

A029993 Squares which are palindromes in base 7.

Original entry on oeis.org

0, 1, 4, 16, 64, 100, 121, 400, 1024, 1600, 2500, 3249, 4096, 6400, 25600, 40000, 118336, 160000, 250000, 302500, 310249, 532900, 1000000, 1893376, 2050624, 3579664, 5769604, 6007401, 6250000, 7573504, 7845601, 11833600

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Squares which are palindromes in base b: A029983 (b=2), A029985 (b=3), A029987 (b=4), A029989 (b=5), A029991 (b=6), this sequence (b=7), A029806 (b=8), A029995 (b=9), A002779 (b=10), A029997 (b=11), A029738 (b=12), A029999 (b=13), A030074 (b=14), A030075 (b=15), A029734 (b=16).

Cf. A029992.

pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[Range[0,3500]^2,pal7Q] (* Harvey P. Dale, Dec 15 2011 *)

a(n) = A029992(n)^2. - Seiichi Manyama, Oct 16 2021

A029995 Squares which are palindromes in base 9.

Original entry on oeis.org

0, 1, 4, 100, 400, 6724, 8281, 10000, 26896, 532900, 672400, 2131600, 43059844, 44129449, 45212176, 53290000, 54479161, 55681444, 172239376, 186104164, 186595600, 203946961, 2921402500, 3486902500, 3583219600

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..52
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029997, A029999, A030074, A030075.

pal9Q[n_]:=Module[{idn9=IntegerDigits[n,9]},idn9==Reverse[idn9]]; Select[ Range[0,60000]^2,pal9Q] (* Harvey P. Dale, Nov 12 2011 *)

cp's OEIS Frontend

A002778 Numbers whose square is a palindrome.

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A033934 a(n) = (10^n + 1)^2.

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A034822 Numbers k such that there are no palindromic squares of length k.

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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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A029734 Palindromic squares in base 16.

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Mathematica

A029738 Palindromic squares in base 12.

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A029806 n in base 8 is a palindromic square.

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Mathematica

A029991 Squares which are palindromes in base 6.

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