A265509 a(n) = largest base-2 palindrome m <= 2n+1 such that every base-2 digit of m is <= the corresponding digit of 2n+1; m is written in base 10.

1, 3, 5, 7, 9, 9, 9, 15, 17, 17, 21, 21, 17, 27, 21, 31, 33, 33, 33, 33, 33, 33, 45, 45, 33, 51, 33, 51, 33, 51, 45, 63, 65, 65, 65, 65, 73, 73, 73, 73, 65, 65, 85, 85, 73, 73, 93, 93, 65, 99, 65, 99, 73, 107, 73, 107, 65, 99, 85, 119, 73, 107, 93, 127, 129, 129, 129, 129, 129, 129, 129, 129, 129, 129, 129, 129, 153

Offset: 0

Author

N. J. A. Sloane, Dec 09 2015

Keywords

• nonn
• base
• look

Comments

A007088(a(n)) = A265510(n). - Reinhard Zumkeller, Dec 11 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Crossrefs

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Cf. A007088, A030308.

Programs

Haskell

a265509 n = a265509_list !! n
a265509_list = f (tail a030308_tabf) [[]] where
   f (bs:_:bss) pss = y : f bss pss' where
     y = foldr (\d v -> 2 * v + d) 0 ys
     (ys:_) = dropWhile (\ps -> not $ and $ zipWith (<=) ps bs) pss'
     pss' = if bs /= reverse bs then pss else bs : pss
-- Reinhard Zumkeller, Dec 11 2015

Maple

ispal := proc(n) # test for base-b palindrome
local L, Ln, i;
global b;
    L := convert(n, base, b);
    Ln := nops(L);
    for i to floor(1/2*Ln) do
        if L[i] <> L[Ln + 1 - i] then return false end if
    end do;
    return true
end proc
# find max pal <= n and in base-b shadow of n, write in base 10
under10:=proc(n) global b;
local t1,t2,i,m,sw1,L2;
if n mod b = 0 then return(0); fi;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then
   t2:=convert(m,base,b);
   L2:=nops(t2);
   sw1:=1;
   for i from 1 to L2 do
      if t2[i] > t1[i] then sw1:=-1; break; fi;
                      od:
   if sw1=1 then return(m); fi;
fi;
                       od;
end proc;
b:=2; [seq(under10(2*n+1),n=0..144)]; # Gives A265509
# find max pal <= n and in base-b shadow of n, write in base b
underb:=proc(n) global b;
local t1,t2,i,m,mb,sw1,L2;
if n mod b = 0 then return(0); fi;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then
   t2:=convert(m,base,b);
   L2:=nops(t2);
   sw1:=1;
   for i from 1 to L2 do
      if t2[i] > t1[i] then sw1:=-1; break; fi;
                      od:
   if sw1=1 then mb:=add(t2[i]*10^(i-1), i=1..L2); return(mb); fi;
fi;
                       od;
end proc;
b:=2; [seq(underb(2*n+1),n=0..144)]; # Gives A265510

Mathematica

A265509 = FromDigits[Min /@ Transpose[{#, Reverse@#}], 2] &@IntegerDigits[2 # + 1, 2] & (* JungHwan Min, Aug 22 2016 *)