# Binary Sequences: Occurrences Of Binary Numbers

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Thu Jul 31 18:43:31 CEST 2008

I have recently posted these three sequences:

%S A143220 1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,0
%N A143220 a(0)=1. For n >=1, a(n) = 1 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 0 otherwise.
%e A143220 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 1101011000111(10101)001. So a(21) = 1.

%Y A143220 A118268,A143221,A143222
%O A143220 0
%K A143220 ,base,more,nonn,

%S A143221 1,0,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0
%N A143221 a(0)=1. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise.
%e A143221 The binary representation of 21 is 10101. This occurs in the concatenation of terms a(0) through a(20) like so: 10010111(10101)00010010. So a(21) = 0.

%Y A143221 A143220,A143222
%O A143221 0
%K A143221 ,base,more,nonn,

%S A143222 0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1
%N A143222 a(0)=0. For n >=1, a(n) = 0 if the binary representation of n occurs at least once in the concatenation of (a(0),a(1),...,a(n-1)). a(n) = 1 otherwise.
%e A143222 The binary representation of 20 is 10100. This occurs in the concatenation of terms a(0) through a(19) like so: 01(10100)1100100111100. So a(20) = 0.

%Y A143222 A143220,A143221
%O A143222 0
%K A143222 ,base,more,nonn,

(Every term of the sequence where a(0)=0 and a(n)=1 if n does occur and a(n)=0 otherwise is trivially 0.)

Let a = sum{k=0 to inf} A143220(k)/2^k,
b =  sum{k=0 to inf} A143221(k)/2^k, and
c =  sum{k=0 to inf} A143222(k)/2^k.

Can a, b, and/or c be connected via any mathematical relations?
Are there closed expressions for a, b, or c?

Maybe the sequences have closed forms for their generating functions, I wonder.
(I am guessing that the GFs with closed forms, if any, would be the ordinary GFs, but I could be wrong.)

Thanks,
Leroy Quet