1,4,9,16,…

You said 25, no? Oh, the presenter turns over the card and it’s a 20. This is the antiharmonic numbers!. Let’s play again

2,4,8

16? Oh, no. It’s 17, the generalized Catalan numbers offset a bit.

1,3,5,7

You said 9. The presenter turns over the card again. It’s 10. This is integers n such that 10^n + 19 is prime

Integer sequences are uncountable. It is important to disabuse ourselves of the notion that there is a canonical sequence associated with just a few terms. Always generate copious terms and check with the Online Encyclopedia of Integer Sequences before jumping to conclusions.