Trouble formatting a sequence

Question

I'm trying to type this problem but I'm having issues as to how I go about the bit within the braces on the right side of the equals sign.

Here's my attempt at formatting this question.
documentclass{article}
usepackage{amsmath, amssymb}

begin{document}

[
left{a_tright}_{t=1}^4 text{ defined recursively by } a_t=
left{(x,y)Bigg|
begin{cases}
x=t 
y=9-t^2
end{cases}
right}
]

end{document}

SebGlav · Accepted Answer

welcome to TeX-SE. Try to fit your cases into a matrix environment, like in the following:
documentclass{article}
usepackage{amsmath, amssymb}

begin{document}

[
left{a_tright}_{t=1}^4 text{ defined recursively by } a_t=
left{(x,y)Bigg|, 
begin{matrix}
x=t 
y=9-t^2
end{matrix}
right}
]

end{document}

egreg · Answer

You can use an array. I present also the version without the inner brace, which I find unnecessary, and my preferred one at the bottom with a single line.
documentclass{article}
usepackage{amsmath, amssymb}

begin{document}

[
{a_t}_{t=1}^4
text{ defined recursively by }
a_t=
left{(x,y);middle|;
left{begin{array}{@{}c@{}}
x=t 
y=9-t^2
end{array}right.
vphantom{Bigg|}
right}
]

[
{a_t}_{t=1}^4
text{ defined recursively by }
a_t=
left{(x,y);middle|;
begin{array}{@{}c@{}}
x=t 
y=9-t^2
end{array}
right}
]

[
{a_t}_{t=1}^4 text{ defined recursively by }
a_t={(x,y) mid x=t, y=9-t^2 }
]

end{document}

Bernard · Answer

I propose this simple code, using an aligned environment, and the set command, as defined by the DeclarePairedDelimiterX command from mathtools:
documentclass{article}
usepackage{mathtools, amssymb}

DeclarePairedDelimiterX{set}[1]{}{setargs{#1}}
NewDocumentCommand{setargs}{>{SplitArgument{1}{;}}m}
{setargsaux#1}
NewDocumentCommand{setargsaux}{mm}
{IfNoValueTF{#2}{#1}{nonscript,#1nonscript;delimsizevertnonscript:allowbreak #2nonscript,}}

begin{document}

[
set{a_t}_{t=1}^4 text{ defined recursively by } a_t=
set[bigg]{(x,y);
begin{aligned}
x & =t [-0.5ex]
y & =9-t^2
end{aligned}}
]

end{document}

Trouble formatting a sequence

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