APL(Dyalog Unicode), ¹⁰9 bytes ^SBCS

⊢∊(+\⍣2⍳) 

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Same as the Vyxal and Uiua answers.

⊢∊(+\⍣2⍳)­⁡​‎‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁢‏⁠‎⁡⁠⁢⁣‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏‏​⁡⁠⁡‌­   ⍳ # ‎⁡range   +\ # ‎⁢cumulative sums   ⍣2 # ‎⁣twice ⊢∊  # ‎⁤input is member 💎 

Created with the help of Luminespire.

APL(Dyalog Unicode), ^{10 9}8 bytes ^SBCS

⊢∊+\⍣2⍤⍳ 

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-1 byte thanks to att.

A tacit function which takes an integer on the right and returns 1 if tetrahedral and 0 if not. Same method as the Vyxal and Uiua answers.

⊢∊+\⍣2⍤⍳­⁡​‎‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁢‏⁠‎⁡⁠⁢⁣‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏‏​⁡⁠⁡‌­ ⍤⍳ # ‎⁡range, then  +\ # ‎⁢cumulative sums, ⍣2 # ‎⁣twice ⊢∊ # ‎⁤input is member 💎 

Created with the help of Luminespire.

APL(Dyalog Unicode), ¹⁰9 bytes ^SBCS

⊢∊(+\⍣2⍳) 

Try it on APLgolf!

Same as the Vyxal and Uiua answers.

APL(Dyalog Unicode), ¹⁰9 bytes ^SBCS

⊢∊(+\⍣2⍳) 

Try it on APLgolf!

Same as the Vyxal and Uiua answers.

⊢∊(+\⍣2⍳)­⁡​‎‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁢‏⁠‎⁡⁠⁢⁣‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏‏​⁡⁠⁡‌­ ⍳ # ‎⁡range +\ # ‎⁢cumulative sums ⍣2 # ‎⁣twice ⊢∊ # ‎⁤input is member 💎 

Created with the help of Luminespire.

APL(Dyalog Unicode), 10 bytes ^SBCS

{⍵∊+\+\⍳⍵} 

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Same as the Vyxal and Uiua answers.

APL(Dyalog Unicode), ¹⁰9 bytes ^SBCS

⊢∊(+\⍣2⍳) 

Try it on APLgolf!

Same as the Vyxal and Uiua answers.