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You Probably Won’t Believe It, But 3(1+φ − φ −1) = 3!

A common formulation for a popular Internet meme that has been in existence since at least 2015 [1, 2, 3] is a variation of the following: Solve carefully: 230 − 220 × 0.5. You probably won’t believe it, but the answer is 5! The apparent contradiction is resolved by recognizing that the actual solution is 5! = 5 × 4 × 3 × 2 × 1 = 120. The meme uses the fact that the symbol for the factorial function coincides with the punctuation mark used to indicate an exclamation (!). What adds to the punchline, however, is that naively giving precedence to the subtraction instead of the multiplication actually results in 5 as a plausible answer. Naturally, one may wonder what other combinations also produce a similar punchline. For a general function a(n), this problem comes down to finding combinations of integers such that