Fibonacci sums the last two. Widen the window to the last three — T(n) = T(n−1) + T(n−2) + T(n−3) — and you get the tribonacci sequence: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44…. Its ratios converge not to φ but to the tribonacci constant ≈ 1.8393 (the root of x³ = x² + x + 1). This is the 3 in of David's puzzle — the recurrence's window opened by one — and it generalizes: sum the last k and the limit climbs toward 2. Fibonacci is just the k=2 member of a whole ladder.
The demo grows the tribonacci sequence — each term the sum of the last three — and shows the ratio climbing past φ toward 1.839: live demo
“φ is the growth rate of ‘add the previous terms’.” — only for two terms. Three gives 1.839, four climbs higher; φ is one rung of a ladder toward 2. cited
One more term in the window, and the golden ratio gives way to 1.839. Fibonacci was only the second rung. Feinberg 1963
On the canonical compiler, the tribonacci sequence 0,0,1,1,2,4,7,13 gives T(7) = 13 — each term the sum of the last three (window 3):