A029966 Palindromic in bases 10 and 11.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 232, 343, 454, 565, 676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854, 4593954, 5643465, 5655565, 5667665

Offset

1,3

Comments

The first 79 terms all have an odd number of decimal digits. Is there a term with an even number of decimal digits? - Robert Israel, Nov 23 2014

Even number of decimal digits is not possible since such decimal palindrome would be divisible by 11, i.e., in base 11 it would have to end and start with 0. - Max Alekseyev, Jun 03 2026

Links

Max Alekseyev, Table of n, a(n) for n = 1..95 (terms for n = 1..79 from Ray Chandler and Robert G. Wilson v)

Patrick De Geest, Palindromic numbers beyond base 10

Maple

N:= 11: # to get all terms with up to N decimal digits
qpali:= proc(k, b) local L; L:= convert(k, base, b); if L = ListTools:-Reverse(L) then k else NULL fi end proc:
digrev:= proc(k, b) local L, n; L:= convert(k, base, b); n:= nops(L); add(L[i]*b^(n-i), i=1..n); end proc:
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(qpali(n*10^m + digrev(n, 10), 11), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(qpali(n*10^(m+1)+y*10^m+digrev(n, 10), 11), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Res;  # Robert Israel, Nov 23 2014

Mathematica

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 12], AppendTo[l, a]], {n, 100000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
(* Alternative: *)
b1=10; b2=11; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10000000}]; lst (* Vincenzo Librandi, Nov 23 2014 *)
(* Alternative: *)
Select[Range[0, 10^5], PalindromeQ[#] && # == IntegerReverse[#, 11] &] (* Robert Price, Nov 09 2019 *)

Prog

(Magma) [n: n in [0..5000000] | Intseq(n) eq Reverse(Intseq(n))and Intseq(n, 11) eq Reverse(Intseq(n, 11))]; // Vincenzo Librandi, Nov 23 2014

Crossrefs

Cf. A046479 (prime terms).

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

Sequence in context: A137667 A117954 A342952 * A219324 A085134 A229761

Adjacent sequences: A029963 A029964 A029965 * A029967 A029968 A029969

Keyword

nonn,base

Author

Patrick De Geest

Status

approved