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A029966 Palindromic in bases 10 and 11.

%I A029966 #41 Aug 11 2024 14:41:30
%S A029966 0,1,2,3,4,5,6,7,8,9,232,343,454,565,676,787,898,909,26962,38183,
%T A029966 40504,49294,52825,63936,75157,2956592,2968692,3262623,3274723,
%U A029966 3286823,3298923,3360633,3372733,4348434,4410144,4422244,4581854
%N A029966 Palindromic in bases 10 and 11.
%C A029966 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
%H A029966 Ray Chandler and Robert G. Wilson v, <a href="/A029966/b029966.txt">Table of n, a(n) for n = 1..79</a>, a(66)-a(76) from Ray Chandler, Oct 31 2014
%H A029966 P. De Geest, <a href="https://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>
%p A029966 N:= 11: # to get all terms with up to N decimal digits
%p A029966 qpali:= proc(k, b) local L; L:= convert(k, base, b); if L = ListTools:-Reverse(L) then k else NULL fi end proc:
%p A029966 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:
%p A029966 Res:= $0..9:
%p A029966 for d from 2 to N do
%p A029966   if d::even then
%p A029966     m:= d/2;
%p A029966     Res:= Res, seq(qpali(n*10^m + digrev(n,10),11), n=10^(m-1)..10^m-1);
%p A029966   else
%p A029966     m:= (d-1)/2;
%p A029966     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);
%p A029966   fi
%p A029966 od:
%p A029966 Res;  # _Robert Israel_, Nov 23 2014
%t A029966 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 *)
%t A029966 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 *)
%t A029966 Select[Range[0, 10^5],
%t A029966 PalindromeQ[#] && # == IntegerReverse[#, 11] &] (* _Robert Price_, Nov 09 2019 *)
%o A029966 (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
%Y A029966 Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029967, A029968, A029969, A029970, A029731, A097855, A099165.
%K A029966 nonn,base
%O A029966 1,3
%A A029966 _Patrick De Geest_