login
Numbers where each binary digit can be paired with a digit of the same value at another position so that two pairs can be nested but cannot otherwise overlap.
1

%I #8 Mar 07 2021 05:53:30

%S 0,3,9,12,15,33,36,39,45,48,51,54,57,60,63,129,132,135,141,144,147,

%T 150,153,156,159,165,177,180,183,189,192,195,198,201,204,207,210,216,

%U 219,222,225,228,231,237,240,243,246,249,252,255,513,516,519,525,528,531

%N Numbers where each binary digit can be paired with a digit of the same value at another position so that two pairs can be nested but cannot otherwise overlap.

%C The term 0 is included by convention (we consider here that it has no digit).

%C This sequence is a binary variant of A333440.

%C Every term belong to A059012.

%C This sequence has connections with A014486; in both sequences digits are balanced in some way.

%H Rémy Sigrist, <a href="/A333441/b333441.txt">Table of n, a(n) for n = 1..8789</a> (terms < 2^16)

%e The first terms, alongside their binary representation with a possible pairing, are:

%e n a(n) bin(a(n))

%e -- ---- ------------

%e 1 0 0

%e 2 3 (11)

%e 3 9 (1(00)1)

%e 4 12 (11)(00)

%e 5 15 (11)(11)

%e 6 33 (1(00)(00)1)

%e 7 36 (1(00)1)(00)

%e 8 39 (1(00)1)(11)

%e 9 45 (1(0(11)0)1)

%e 10 48 (11)(00)(00)

%e 11 51 (11)(00)(11)

%e 12 54 (11)(0(11)0)

%e 13 57 (11)(1(00)1)

%e 14 60 (11)(11)(00)

%e 15 63 (11)(11)(11)

%o (PARI) is(n, base=2) = { my (u=0, s=0); while (n, my (d=n%base); if (u && s%base==d, u--; s\=base, u++; s=s*base+d); n\=base); u==0 }

%Y Cf. A014486, A059012, A333440.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Mar 21 2020