A215244 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of palindromes.

1, 1, 1, 2, 2, 2, 2, 4, 4, 3, 3, 3, 4, 3, 4, 8, 8, 5, 4, 5, 5, 5, 4, 6, 8, 5, 5, 6, 8, 6, 8, 16, 16, 9, 7, 9, 8, 6, 6, 10, 10, 6, 8, 8, 7, 7, 7, 12, 16, 9, 7, 10, 8, 8, 8, 11, 16, 10, 10, 11, 16, 12, 16, 32, 32, 17, 13, 17, 13, 11, 11, 18, 15, 11, 10, 10, 12, 9, 11, 20, 20, 11, 10, 11, 13, 13, 10, 16, 14, 9, 10, 12, 13, 11, 13, 24, 32, 17, 13, 18, 14, 11, 12, 19, 16, 10, 13

Offset: 0

Author

N. J. A. Sloane, Aug 07 2012

Keywords

• nonn
• base
• look

Comments

"Product" is used here is the usual sense of combinatorics on words.

The sequence may be arranged into a triangle in which row k contains the 2^k terms [a(2^k), ..., a(2^(k+1)-1)]: (1), (1), (1,2), (2,2,2,4), ... The row sums of this triangle appear to be A063782. - R. J. Mathar, Aug 09 2012. I have a proof that A063782 does indeed give the row sums. - N. J. A. Sloane, Aug 10 2012

a(A220654(n)) = n and a(m) < n for m < A220654(n). - Reinhard Zumkeller, Dec 17 2012

Examples

			a(4)=2 since 4 = 100, and 100 can be written as either 1.0.0 or 1.00.
a(5)=2 from 1.0.1, 101.
a(10)=3 from 1.0.1.0, 1.010, 101.1
Written as a triangle, the sequence is:
1,
1,
1, 2,
2, 2, 2, 4,
4, 3, 3, 3, 4, 3, 4, 8,
8, 5, 4, 5, 5, 5, 4, 6, 8, 5, 5, 6, 8, 6, 8, 16,
16, 9, 7, 9, 8, 6, 6, 10, 10, 6, 8, 8, 7, 7, 7, 12, 16, 9, 7, 10, 8, 8, 8, 11, 16, 10, 10, 11, 16, 12, 16, 32,
...
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to palindromes

Crossrefs

Cf. A050430, A215245, A215246, A215250, A215253, A215254, A063782.

Cf. A206925, A030308.

Programs

Haskell

import Data.Map (Map, singleton, (!), insert)
import Data.List (inits, tails)
newtype Bin = Bin [Int] deriving (Eq, Show, Read)
instance Ord Bin where
   Bin us <= Bin vs | length us == length vs = us <= vs
                    | otherwise              = length us <= length vs
a215244 n = a215244_list !! n
a215244_list = 1 : f [1] (singleton (Bin [0]) 1) where
   f bs m | last bs == 1 = y : f (succ bs) (insert (Bin bs) y m)
          | otherwise    = f (succ bs) (insert (Bin bs) y m) where
     y = fromEnum (pal bs) +
         sum (zipWith (\us vs -> if pal us then m ! Bin vs else 0)
                      (init $ drop 1 $ inits bs) (drop 1 $ tails bs))
     pal ds = reverse ds == ds
     succ [] = [0]; succ (0:ds) = 1 : ds; succ (1:ds) = 0 : succ ds
-- Reinhard Zumkeller, Dec 17 2012

Maple

isPal := proc(L)
    local d ;
    for d from 1 to nops(L)/2 do
        if op(d,L) <> op(-d,L) then
            return false;
        end if;
    end do:
    return true;
end proc:
A215244L := proc(L)
    local a,c;
    a := 0 ;
    if isPal(L) then
        a := 1;
    end if;
    for c from 1 to nops(L)-1 do
        if isPal( [op(1..c,L)] ) then
            a := a+procname([op(c+1..nops(L),L)]) ;
        end if;
    end do:
    return a;
end proc:
A215244 := proc(n)
    if n = 1 then
        1;
    else
        convert(n,base,2) ;
        A215244L(%) ;
    end if;
end proc: # R. J. Mathar, Aug 07 2012
# Caution: the last procedure applies A215244L to the reverse of the binary expansion of n, which is perfectly OK but might cause a problem if the procedure was used in a different problem. - N. J. A. Sloane, Aug 11 2012

Mathematica

palQ[L_] := SameQ[L, Reverse[L]];
b[L_] := b[L] = Module[{a = palQ[L] // Boole, c}, For[c = 1, c < Length[L], c++, If[palQ[L[[;;c]]], a = a + b[L[[c+1;;]]]]]; a];
a[n_] := If[n == 1, 1, b[IntegerDigits[n, 2]]];
a /@ Range[0, 106] (* Jean-François Alcover, Oct 27 2019 *)