A215254 -id:A215254 - cp's OEIS frontend

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

N. J. A. Sloane, Aug 07 2012

"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

			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,
...

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

Cf. A206925, A030308.

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

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

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 *)

A215246 a(n) = maximal value of k such that A215244(k) = n.

Original entry on oeis.org

2, 6, 13, 22, 26, 41, 50, 54, 89, 105, 101, 109, 178, 166, 118, 185, 217, 218, 201, 357, 361, 358, 422, 329, 434, 425, 429, 418, 617, 441, 666, 717, 845, 722, 745, 741, 718, 873, 869, 841, 846, 1430, 882, 844, 866, 934, 946, 1446, 1434, 885, 1690, 1625, 1445, 1686, 949, 1714, 1497

Offset: 1

N. J. A. Sloane, Aug 07 2012

			A215244(k) = 4 for k = 7,8,12,14,18,22, so a(4) = 22.

N. J. A. Sloane, Table of n, a(n) for n = 1..307

Cf. A215244, A215245, A215253, A215254.

# Since A214245(12)=95, the following program gives correct values for a(n) for n <= 94.
A215246:=proc(M) local lis,n,p,i;
lis:=array(1..1024);
for n from 1 to M do
p:=A215244(n);
if p <= 1024 then lis[p]:=n; fi;
                  od;
[seq(lis[i],i=1..94)];
end proc;
A215246(2^12 - 1);

A215245 a(n) = minimal value of A215244(k) for 2^n <= k < 2^(n+1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 20, 29, 42, 65, 95, 136, 212, 308, 444, 687, 1005, 1439, 2242, 3257, 4696, 7266, 10629, 15219

Offset: 0

N. J. A. Sloane, Aug 07 2012

The initial terms roughly satisfy a(n) approx.= a(n-1)+a(n-3), which leads to the guess that perhaps a(n) ~ 1.4655^n, from the real zero of x^3-x-1. - N. J. A. Sloane, Aug 08 2012

			The values of A215244(k) for k=8 through 15 are (4, 3, 3, 3, 4, 3, 4, 8), with minimal value a(3) = 3.

Giovanni Resta, Examples of words attaining the minimal value for n = 0..25

Cf. A215244, A215246, A215253, A215254. A215255 gives an upper bound.

A215245 := proc(n)
    local a,k ;
    a := A215244(2^n) ;
    for k from 2^n+1 to 2^(n+1)-1 do
        a := min(a,A215244(k)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 07 2012

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];
a215244[n_] := If[n == 1, 1, b[IntegerDigits[n, 2]]];
a215245[n_] := Module[{a, k}, a = a215244[2^n]; For[k = 2^n+1, k <= 2^(n+1) - 1, k++, a = Min[a, a215244[k]]]; a];
a215245 /@ Range[0, 20] (* Jean-François Alcover, Oct 28 2019 *)

a(10)-a(13) from R. J. Mathar, Aug 07 2012
a(14)-a(17) from N. J. A. Sloane, Aug 08 2012, using Mathar's Maple code.
a(18)-a(25) from Giovanni Resta, Mar 19 2013

A215255 Let S be the binary string consisting of the first n digits of (100101)*; a(n) = number of ways of writing S as a product of palindromes.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 10, 13, 23, 29, 42, 65, 107, 136, 243, 308, 444, 687, 1131, 1439, 2570, 3257, 4696, 7266, 11962, 15219, 27181, 34447, 49666, 76847, 126513, 160960, 287473, 364320, 525280, 812753, 1338033, 1702353, 3040386, 3853139

Offset: 0

N. J. A. Sloane, Aug 14 2012

If S is the binary representation of the decimal number N, then a(n) = A215244(N).

a(n) is an upper bound for A215245(n), which might be tight infinitely often.

Cf. A215244, A215245, A215246, A215254.

Recurrence: For n >= 4, a(n) = a(n-1)+a(n-d), where d = [3,2,4,2,4,3] according as n == [0,1,2,3,4,5] mod 6; initial conditions a(0)=a(1)=a(2)=1, a(3)=2.
G.f.: (x^17+x^14+x^12+5*x^11+2*x^10-x^9+3*x^8+3*x^7+6*x^5+4*x^4+3*x^3+2*x^2+x+1)/(1-10*x^6-6*x^12-x^18).
a(n) ~ C * D^n, where D = 1.4815692... and C depends on n mod 6 (approximate values of C are [0.580722..., 0.6452899..., 0.554135..., 0.667994..., 0.571395..., 0.556061...], respectively).

cp's OEIS Frontend

A215253 a(n) = decimal equivalent of A215254(n).

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A215246 a(n) = maximal value of k such that A215244(k) = n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A215245 a(n) = minimal value of A215244(k) for 2^n <= k < 2^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A215255 Let S be the binary string consisting of the first n digits of (100101)*; a(n) = number of ways of writing S as a product of palindromes.

Views

Author

Keywords

Comments

Crossrefs

Formula