The lowest initial numbers in a Fibonacci-like sequence
APL (Dyalog), 75 71 64 59 53 48 44 43 bytes

2 bytes saved thanks to @Adám

12 bytes saved thanks to @ngn



⊃o/⍨k∊¨+\∘⌽⍣{k≤⊃⍺}¨o←a/⍨&lt;/¨a←,⍉|-\¨⍳2⍴1+k←⎕


Try it online!

Uses ⎕IO←0.

How? This had gone real nuts.

k←⎕ - assign input to k

⍳2⍴1+k←⎕ - Cartesian product of the range 0 to k with itself

|-\¨ - substract each right pair element from the left, and get absolute values

a←,⍉ - transpose, flatten and assign to a

o←a/⍨&lt;/¨a - keep only pairs where the left element is smaller than the right, and assign to o

o now contains list of all pairs with a &lt; b, ordered by their arithematical mean

+\∘⌽⍣{k≤⊃⍺}¨o - for each pair in o, apply fibonacci (reverse the pair and cumsum) until k or higher term is reached

k∊¨ - then decide if k is this last term (meaning it is contained in the sequence)

o/⍨ - and keep pairs in o where the previous check applies

⊃ - return the first result.