A214218 List of words over {1,2} with equal numbers of 1's and 2's.

12, 21, 1122, 1212, 1221, 2112, 2121, 2211, 111222, 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111, 11112222, 11121222, 11122122, 11122212, 11122221

Offset: 1

Author

N. J. A. Sloane, Jul 18 2012

Keywords

• nonn
• base

Comments

Of course the empty word also has this property.

All of these, interpreted as decimal integers are divisible by 3, because each pair of "1" and "2" contributes a digital sum of 3, hence the total is divisible by 3. Is there a semiprime in the sequence after 21? - Jonathan Vos Post, Jul 18 2012

The semiprime subsequence contains 21, 11222121, 12122211, 21221121, 22211121, 22212111, and continues with 14 10-digit entries etc. - R. J. Mathar, Jul 19 2012

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 2.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A007931, A111066.

Programs

Maple

sort([seq(seq((10^(2*d)-1)/9+add(10^i,i=s),s=combinat:-choose([$0..(2*d-1)],d)),d=1..4)]); # Robert Israel, Jan 02 2018

Mathematica

Sort[FromDigits/@Flatten[Table[Permutations[PadRight[{},2n,{1,2}]],{n,3}],1]] (* Harvey P. Dale, Aug 30 2016 *)

Python

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations as mp
def agen():
    for d in count(2, 2):
        for s in mp("1"*(d//2) + "2"*(d//2), d):
            yield int("".join(s))
print(list(islice(agen(), 33))) # Michael S. Branicky, Dec 21 2021