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Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group 2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).

P.S. This is really a comment; can somebody with privilege demote it to the appropriate place, please?

Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group 2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).

Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group N/2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).

P.S. This is really a comment; can somebody with privilege demote it to the appropriate place, please?

Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group 2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).

P.S. This is really a comment; can somebody with privilege demote it to the appropriate place, please?

If the numbers with odd number of 1s are viewed as a set (as opposed to sequence), then the development follows the classic theory of formal languages. The set of numbers with odd number of 1s can be defined as formal language Z by the following equations in Kleene algebra:

X = "" + "0" X + X "0" 
Y = "" + X "1" Y "1" X
Z = X "1" Y

where the "" is the empty string, "0" and "1" are letters of the alphabet, the string concatenation is multiplicative operation, and the + is set union. The empty string "" is multiplicative unit, however its standard notation 1 would interfere with the alphabet.

A set of numbers with odd number of 1s is a regular language which implies efficient computability. Kleene algebras are idempotent semirings.

Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group N/2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  
   ^  ^     ^           ^   ^           ^       ^   ^       ^           ^       ^   
----  ----  -----   -----   -----  ------  ------   -----   -----  ------  ------

each pair [2n,2n+1] is a group element is, so that we have

[a(6),a(7)] + [a(10),a(11)] = [a(16),a(17)]

and yet there seems to be no relation among a(6), a(10) and a(16).

P.S. This is really a comment; can somebody with privilege demote it to the appropriate place, please?