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

1,1

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

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 *)

Prog

(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

Crossrefs

Subsequence of A007931, A111066.

Sequence in context: A162391 A114015 A065439 * A323084 A323083 A212958

Adjacent sequences: A214215 A214216 A214217 * A214219 A214220 A214221

Keyword

nonn,base

Author

N. J. A. Sloane, Jul 18 2012

Status

approved