Desmos, 51 50 45 bytes

o→\left\{i=0:[0],join(1,o!)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

EDIT: a surprising -1, apparently → parses properly as an action! Unfortunately, it still costs 3 bytes, which is the same as \to, BUT it allows me to change \to join to →join since the parser gets confused by \tojoin

EDIT: -5 thanks to zygan on discord who showed me that an action itself can be piecewise, and also me realizing that o! is shorter than o^o and doesn't even have to rely on wacky edge case behavior lol

Old 50 byte solution:

\left\{i=0:o→[0],o→join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Old 51 byte solution:

\left\{i=0:o\to[0],o\to join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos, 51 50 45 bytes

o→\left\{i=0:[0],join(1,o!)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

Addendum: at the very least, this output method is accepted by default. Not sure about the input.

EDIT: a surprising -1, apparently → parses properly as an action! Unfortunately, it still costs 3 bytes, which is the same as \to, BUT it allows me to change \to join to →join since the parser gets confused by \tojoin

EDIT: -5 thanks to zygan on discord who showed me that an action itself can be piecewise, and also me realizing that o! is shorter than o^o and doesn't even have to rely on wacky edge case behavior lol

Desmos, 51 50 bytes

\left\{i=0:o→[0],o→join(1,o^o)\right\}
i=0
o=0

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

EDIT: a surprising -1, apparently → parses properly as an action! Unfortunately, it still costs 3 bytes, which is the same as \to, BUT it allows me to change \to join to →join since the parser gets confused by \tojoin

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Old 51 byte solution:

\left\{i=0:o\to[0],o\to join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos, 51 50 45 bytes

o→\left\{i=0:[0],join(1,o!)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

EDIT: a surprising -1, apparently → parses properly as an action! Unfortunately, it still costs 3 bytes, which is the same as \to, BUT it allows me to change \to join to →join since the parser gets confused by \tojoin

EDIT: -5 thanks to zygan on discord who showed me that an action itself can be piecewise, and also me realizing that o! is shorter than o^o and doesn't even have to rely on wacky edge case behavior lol

Old 50 byte solution:

\left\{i=0:o→[0],o→join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Old 51 byte solution:

\left\{i=0:o\to[0],o\to join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos, 51 bytes

\left\{i=0:o\to[0],o\to join(1,o^o)\right\}
i=0
o=0

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Desmos, 51 50 bytes

\left\{i=0:o→[0],o→join(1,o^o)\right\}
i=0
o=0

Desmos doesn't exactly have standard I/O support, but this should get the point across. Input is the i variable, output is the o variable. Each line is a separate expression, and the first expression needs to be in the ticker, as shown in the TIO.

EDIT: a surprising -1, apparently → parses properly as an action! Unfortunately, it still costs 3 bytes, which is the same as \to, BUT it allows me to change \to join to →join since the parser gets confused by \tojoin

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)

Old 51 byte solution:

\left\{i=0:o\to[0],o\to join(1,o^o)\right\}
i=0
o=0

Try It Online! (Click the play button on the ticker up top (metronome icon) to run the program)