A006995 -id:A006995 - cp's OEIS frontend

A014192 Palindromes in base 4 (written in base 10).

0, 1, 2, 3, 5, 10, 15, 17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255, 257, 273, 289, 305, 325, 341, 357, 373, 393, 409, 425, 441, 461, 477, 493, 509, 514, 530, 546, 562, 582, 598, 614, 630, 650, 666

Offset: 1

N. J. A. Sloane

Rajasekaran, Shallit, & Smith prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of palindromes: an approach via automata, arXiv:1706.10206 [cs.FL], 2017.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

[n: n in [0..800] | Intseq(n, 4) eq Reverse(Intseq(n, 4))]; // Vincenzo Librandi, Sep 09 2015

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,4], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal4Q[n_]:=Module[{c=IntegerDigits[n,4]},c==Reverse[c]]; Select[Range[ 0,700],pal4Q] (* Harvey P. Dale, Jul 21 2020 *)

ispal(n,b=4)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def A014192(n):
    if n == 1: return 0
    y = (x:=1<<(n.bit_length()-2&-2))<<2
    return (c:=n-x)*x+int(digits(c,4)[-2::-1]or'0',4) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 2.7857715... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

More terms from Patrick De Geest

A057148 Palindromes only using 0 and 1 (i.e., base-2 palindromes).

Original entry on oeis.org

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, 101101, 110011, 111111, 1000001, 1001001, 1010101, 1011101, 1100011, 1101011, 1110111, 1111111, 10000001, 10011001, 10100101, 10111101, 11000011, 11011011, 11100111, 11111111, 100000001

Offset: 1

Henry Bottomley, Aug 14 2000

For each term having fewer than 10 digits, the square will also be a palindrome. - Dmitry Kamenetsky, Oct 21 2008

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A006995 for sequence translated from binary to decimal. A016116 for number of terms of sequence with n+1 binary digits (0 taken to have no digits).

Cf. A002113, A118594, A118595, A118596, A118597, A118598, A118599, A118600.

(* get NextPalindrome from A029965 *)
Select[ NestList[ NextPalindrome, 0, 11110], Max(AT) IntegerDigits(AT)# < 2 &] (* Robert G. Wilson v *)
Select[FromDigits/@Tuples[{0,1},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1]
    for d in count(2):
        for rest in product("01", repeat=d//2-1):
            left = "1" + "".join(rest)
            for mid in [[""], ["0", "1"]][d%2]:
                yield int(left + mid + left[::-1])
print(list(islice(agen(), 32))) # Michael S. Branicky, Mar 29 2022

def A057148(n):
    if n == 1: return 0
    a = 1<Chai Wah Wu, Jun 10 2024

[int(n.binary()) for n in (0..220) if Word(n.digits(2)).is_palindrome()] # Peter Luschny, Sep 13 2018

A060792 Numbers that are palindromic in bases 2 and 3.

Original entry on oeis.org

0, 1, 6643, 1422773, 5415589, 90396755477, 381920985378904469, 1922624336133018996235, 2004595370006815987563563, 8022581057533823761829436662099, 392629621582222667733213907054116073, 32456836304775204439912231201966254787, 428027336071597254024922793107218595973

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

a(18) (if it exists) is greater than 3^93. - Ilya Nikulshin, Feb 22 2016

			6643 is a term: since 6643 = 1100111110011_2 = 100010001_3.
1422773 is a term: 1422773 = 101011011010110110101_2 = 2200021200022_3. - _Vladimir Joseph Stephan Orlovsky_, Sep 19 2009

Ilya Nikulshin, Table of n, a(n) for n = 1..17 (terms a(11)..a(15) from Alan Grimes and _Keith F. Lynch_; a(16) from _Japheth Lim_)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.
Alan Grimes, Keith F. Lynch, et al., Palindrome numbers.
Keith F. Lynch, How I found big dual palindromes.

a(3) = A048268(2) = A056749(3).

Intersection of A006995 and A014190.

[n: n in [0..2*10^7] | Intseq(n, 3) eq Reverse(Intseq(n, 3))and Intseq(n, 2) eq Reverse(Intseq(n, 2))]; // Vincenzo Librandi, Feb 24 2016

pal2Q[n_Integer] := IntegerDigits[n, 2] == Reverse[IntegerDigits[n, 2]]; pal3Q[n_Integer] := IntegerDigits[n, 3] == Reverse[IntegerDigits[n, 3]]; A060792 = {}; Do[If[pal2Q[n] && pal3Q[n], AppendTo[A060792, n]], {n, 12!}]; A060792 (* Vladimir Joseph Stephan Orlovsky, Sep 19 2009 *)
b1=2; b2=3; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 2 10^7}]; lst (* Vincenzo Librandi, Feb 24 2016 *)

ispal(n,b)=my(d=digits(n,b)); d==Vecrev(d)
is(n)=ispal(n,2)&&ispal(n,3) \\ Charles R Greathouse IV, Jun 17 2014

from itertools import chain
from gmpy2 import digits, mpz
A060792 = [int(n,2) for n in chain(map(lambda x:bin(x)[2:]+bin(x)[2:][::-1],range(1,2**16)),map(lambda x:bin(x)[2:]+bin(x)[2:][-2::-1], range(1,2**16))) if mpz(int(n,2)).digits(3) == mpz(int(n,2)).digits(3)[::-1]] # Chai Wah Wu, Aug 12 2014

a(7) found by François Boisson, using a Caml program running on an AMD-64 machine. - Bruno Petazzoni, program co-author, Jan 31 2006
a(8) from the same source, May 26 2006
a(9) from Alan Grimes, Dec 16 2013
a(10) from Keith F. Lynch, Jan 07 2014
Term 0 prepended by Robert G. Wilson v, Oct 08 2014
a(11)-a(15) (from Alan Grimes and Keith F. Lynch) added by Japheth Lim, Jan 30 2014

A094202 Integers k whose Zeckendorf representation A014417(k) is palindromic.

Original entry on oeis.org

0, 1, 4, 6, 9, 12, 14, 22, 27, 33, 35, 51, 56, 64, 74, 80, 88, 90, 116, 127, 145, 158, 174, 184, 197, 203, 216, 232, 234, 276, 294, 326, 368, 378, 399, 425, 441, 462, 472, 493, 519, 525, 546, 572, 588, 609, 611, 679, 708, 760, 828, 847, 915, 944, 988, 1022, 1064, 1090

Offset: 1

Ron Knott, May 25 2004

			Fibonacci base columns are ...,8,5,3,2,1 with column entries 0 or 1 and no two consecutive ones (the Zeckendorf representation) so that each n has a unique representation.
12 is in the sequence because 12 = 8 + 3 + 1 = 10101 base Fib; 14 = 13 + 1 = 100001 base Fib.

C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin vol. 29, 1952, pages 190-195.
E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Société Royale des Sciences de Liège vol. 41 (1972) pages 179-182.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 129 terms from Indranil Ghosh)
Ron Knott, Fibonacci Bases.

Cf. A014417, A035517.

Gives the positions of zeros in A095734. Subsets: A095730, A048757. A006995 gives the integers whose binary expansion is palindromic.

zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[ fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; a = {}; Do[z = zeck[n]; If[ FromDigits[ Reverse[ IntegerDigits[z]]] == z, AppendTo[a, n]], {n, 1123}]; a (* Robert G. Wilson v, May 29 2004 *)
mirror[dig_, s_] := Join[dig, s, Reverse[dig]]; select[v_, mid_] := Select[v, Length[#] == 0 || Last[#] != mid &]; fib[dig_] := Plus @@ (dig * Fibonacci[Range[2, Length[dig] + 1]]); pals = Rest[IntegerDigits /@ FromDigits /@ Select[Tuples[{0, 1}, 7], SequenceCount[#, {1, 1}] == 0 &]]; Union@Join[{0, 1}, fib /@ Join[mirror[#, {}] & /@ (select[pals, 1]), mirror[#, {1}] & /@ (select[pals, 1]), mirror[#, {0}] & /@ pals]] (* Amiram Eldar, Jan 11 2020 *)

from sympy import fibonacci
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n):
    x=str(a(n))
    return x==x[::-1]
print([n for n in range(1101) if ok(n)]) # Indranil Ghosh, Jun 07 2017

More terms from Robert G. Wilson v, May 28 2004

Offset changed to 1 by Alois P. Heinz, Aug 02 2017

A029952 Palindromic in base 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 18, 24, 26, 31, 36, 41, 46, 52, 57, 62, 67, 72, 78, 83, 88, 93, 98, 104, 109, 114, 119, 124, 126, 156, 186, 216, 246, 252, 282, 312, 342, 372, 378, 408, 438, 468, 498, 504, 534, 564, 594, 624, 626, 651, 676, 701, 726, 756, 781, 806, 831

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

Cf. A261917, A261918.

[n: n in [0..900] | Intseq(n, 5) eq Reverse(Intseq(n, 5))]; // Vincenzo Librandi, Sep 09 2015

# test for palindrome in base b, from N. J. A. Sloane, Sep 13 2015
b:=5;
ispal := proc(n) global b; local t1,t2,i;
if n <= b-1 then return(1); fi;
t1:=convert(n,base,b); t2:=nops(t1);
for i from 1 to floor(t2/2) do
if t1[i] <> t1[t2+1-1] then return(-1); fi;
od: return(1); end;
lis:=[]; for n from 0 to 100 do if ispal(n) = 1 then lis:=[op(lis),n]; fi; od: lis;

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,5], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range[0,1000],IntegerDigits[#,5]==Reverse[IntegerDigits[#,5]]&] (* Harvey P. Dale, Oct 24 2020 *)

ispal(n,b=5)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def A029952(n):
    if n == 1: return 0
    y = 5*(x:=5**(len(digits(n>>1,5))-1))
    return int((c:=n-x)*x+int(digits(c,5)[-2::-1]or'0',5) if nChai Wah Wu, Jun 13 2024

Sum_{n>=2} 1/a(n) = 2.9200482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029954 Palindromic in base 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 50, 57, 64, 71, 78, 85, 92, 100, 107, 114, 121, 128, 135, 142, 150, 157, 164, 171, 178, 185, 192, 200, 207, 214, 221, 228, 235, 242, 250, 257, 264, 271, 278, 285, 292, 300, 307, 314, 321, 328, 335, 342, 344, 400, 456

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,7], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[ Range[0,500],pal7Q] (* Harvey P. Dale, Jul 30 2015 *)

ispal(n,b=7)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[-2::-1],b)
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[::-1],b)
A029954_list = list(palQgen(4,7)) # Chai Wah Wu, Dec 01 2014

from gmpy2 import digits
from sympy import integer_log
def A029954(n):
    if n == 1: return 0
    y = 7*(x:=7**integer_log(n>>1,7)[0])
    return int((c:=n-x)*x+int(digits(c,7)[-2::-1]or'0',7) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.1313768... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A057890 In base 2, either a palindrome or becomes a palindrome if trailing 0's are omitted.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 27, 28, 30, 31, 32, 33, 34, 36, 40, 42, 45, 48, 51, 54, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 90, 93, 96, 99, 102, 107, 108, 112, 119, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146

Offset: 1

Marc LeBrun, Sep 25 2000

Symmetric bit strings (bit-reverse palindromes), including as many leading as trailing zeros.
Fixed points of A057889, complement of A057891
n such that A000265(n) is in A006995. - Robert Israel, Jun 07 2016

			10 is included, since 01010 is a palindrome, but 11 is not because 1011 is not.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of Palindromes: an Approach via Nested-Word Automata, preprint arXiv:1706.10206 [cs.FL], June 30 2017.

Cf. A030101, A000265, A006519, A006995, A057889, A057891, A061917, A273245, A273329, A272670.

a057890 n = a057890_list !! (n-1)
a057890_list = 0 : filter ((== 1) . a178225 . a000265) [1..]
-- Reinhard Zumkeller, Oct 21 2011

dmax:= 10: # to get all terms < 2^dmax
revdigs:= proc(n)
  local L, Ln, i;
  L:= convert(n, base, 2);
  Ln:= nops(L);
  add(L[i]*2^(Ln-i), i=1..Ln);
end proc;
P[0]:= {0}:
P[1]:= {1}:
for d from 2 to dmax do
  if d::even then
    P[d]:= { seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)}
  else
    m:= (d-1)/2;
    B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)};
    P[d]:= B union map(`+`, B, 2^m)
  fi
od:
A:= `union`(seq(seq(map(`*`,P[d],2^k),k=0..dmax-d),d=0..dmax)):
sort(convert(A,list)); # Robert Israel, Jun 07 2016

PaleQ[n_Integer, base_Integer] := Module[{idn, trim = n/base^IntegerExponent[n, base]}, idn = IntegerDigits[trim, base]; idn == Reverse[idn]]; Select[Range[0, 150], PaleQ[#, 2] &] (* Lei Zhou, Dec 13 2013 *)
pal2Q[n_]:=Module[{id=Drop[IntegerDigits[n,2],-IntegerExponent[n,2]]},id==Reverse[id]]; Join[{0},Select[Range[200],pal2Q]] (* Harvey P. Dale, Feb 26 2015 *)
A057890Q = If[# > 0 && EvenQ@#, #0[#/2], # == #~IntegerReverse~2] &; Select[0~Range~146, A057890Q] (* JungHwan Min, Mar 29 2017 *)
Select[Range[0, 200], PalindromeQ[IntegerDigits[#, 2] /. {b__, 0..} -> {b} ]&] (* Jean-François Alcover, Sep 18 2018 *)

bitrev(n) = subst(Pol(Vecrev(binary(n>>valuation(n,2))), 'x), 'x, 2);
is(n) = my(x = n >> valuation(n,2)); x == bitrev(x);
concat(0, select(is,vector(147,n,n)))  \\ Gheorghe Coserea, Jun 07 2016

is(n)=n==0 || Vecrev(n=binary(n>>valuation(n,2)))==n \\ Charles R Greathouse IV, Aug 25 2016

A057890 = [n for n in range(10**6) if bin(n)[2:].rstrip('0') == bin(n)[2:].rstrip('0')[::-1]] # Chai Wah Wu, Aug 12 2014

A030101(A030101(n)) = A030101(n). - David W. Wilson, Jun 09 2009, Jun 18 2009
A178225(A000265(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2011
a(7*2^n-4*n-4) = 4^n + 1, a(10*2^n-4*n-6) = 2*4^n + 1. - Gheorghe Coserea, Apr 05 2017

A029953 Palindromic in base 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 37, 43, 49, 55, 61, 67, 74, 80, 86, 92, 98, 104, 111, 117, 123, 129, 135, 141, 148, 154, 160, 166, 172, 178, 185, 191, 197, 203, 209, 215, 217, 259, 301, 343, 385, 427, 434, 476, 518, 560, 602, 644, 651, 693, 735, 777, 819

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

[n: n in [0..900] | Intseq(n, 6) eq Reverse(Intseq(n, 6))]; // Vincenzo Librandi, Sep 09 2015

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,6], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

ispal(n,b=6)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
from sympy import integer_log
def A029953(n):
    if n == 1: return 0
    y = 6*(x:=6**integer_log(n>>1,6)[0])
    return int((c:=n-x)*x+int(digits(c,6)[-2::-1]or'0',6) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.03303318... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A003166 Numbers whose square in base 2 is a palindrome.

Original entry on oeis.org

0, 1, 3, 4523, 11991, 18197, 141683, 1092489, 3168099, 6435309, 12489657, 17906499, 68301841, 295742437, 390117873, 542959199, 4770504939, 17360493407, 73798050723, 101657343993, 107137400475, 202491428745, 1615452642807, 4902182461643, 9274278357017, 12863364360297

Offset: 1

N. J. A. Sloane, R. H. Hardin

Numbers k such that k^2 is in A006995.

The only palindromes in this sequence are 0, 1, and 3. See AMM problem 11922. - Max Alekseyev, Oct 22 2022

			3^2 = 9 = 1001_2, a palindrome.
4523^2 = 20457529 = 1001110000010100000111001_2.

G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Don Knuth, Table of n, a(n) for n = 1..50 [This table extends earlier work of Gus Simmons, Jon Schoenfield, Don Knuth, and Michael Coriand]
M. A. Alekseyev, Problem 11922. American Mathematical Monthly 123:7 (2016), 722.
Patrick De Geest, Palindromic Squares in bases 2 to 17
Carlos Rivera, Problem 89. Palindromic binary expression of primes squared, The Prime Puzzles & Problems Connection.
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778 (base 10 analog), A029983 (the actual squares). In binary: A262595, A262596.

Cf. A006995.

Do[c = RealDigits[n^2, 2][[1]]; If[c == Reverse[c], Print[n]], {n, 0, 10^9}]

is(n)=my(b=binary(n^2)); b==Vecrev(b) \\ Charles R Greathouse IV, Feb 07 2017

from itertools import count, islice
def A003166_gen(): # generator of terms
    return filter(lambda k: (s:=bin(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A003166_list = list(islice(A003166_gen(),10)) # Chai Wah Wu, Jun 23 2022

a(16) = 4770504939 found by Patrick De Geest, May 15 1999
a(17)-a(31) from Jon E. Schoenfield, May 08 2009
a(32) = 285000288617375,
a(33) = 301429589329949,
a(34) = 1178448744881657 from Don Knuth, Jan 28 2013 [who doublechecked the previous results and searched up to 2^104]

A097856 Base 10 numbers that are palindromic in bases 2 and 4.

Original entry on oeis.org

0, 1, 3, 5, 15, 17, 21, 51, 63, 65, 85, 195, 255, 257, 273, 325, 341, 771, 819, 975, 1023, 1025, 1105, 1285, 1365, 3075, 3315, 3855, 4095, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 12291, 12483, 13107, 13299, 15375, 15567, 16191, 16383, 16385, 16705

Offset: 1

Cino Hilliard, Aug 31 2004

Intersection of A014192 and A006995. - Michel Marcus, Oct 11 2014

			255 base 2 = 11111111 and 255 base 4 = 3333.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A014192 (base 4), A006995 (base 2).

cp's OEIS Frontend

A014192 Palindromes in base 4 (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A057148 Palindromes only using 0 and 1 (i.e., base-2 palindromes).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Sage

A060792 Numbers that are palindromic in bases 2 and 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

A094202 Integers k whose Zeckendorf representation A014417(k) is palindromic.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A029952 Palindromic in base 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A029954 Palindromic in base 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula