Am I a Special N-bonacci Number?

05AB1E, 18 bytes

[DR²£O©‚˜³®>‹#]³QZ

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Uses: [X,Y], N, A

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I feel like some unintended functionality made that harder than it needed to be.

There's no greater-than-or-equal-to, never noticed that before.

And #³ didn't work, and required a ], for +1 bytes #]³.

Jelly, 12 bytes

ḣ⁴S;µṀ<⁵µ¿⁵e

A full program taking [X,Y], N, A.

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How?

ḣ⁴S;µṀ<⁵µ¿⁵e - Main link (monadic): [X,Y]
    µ   µ¿   - while:
     Ṁ       -   maximum value of the list
       ⁵     -   5th command line argument (3rd input) = A
      <      -   less than?
             - ...do:
 ⁴           -   4th command line argument (2nd input) = N
ḣ            -   head (get the first N (or less) items from the list)
  S          -   sum
   ;         -   concatenate (add the result to the front of the list)
          ⁵  - 5th command line argument (3rd input) = A
           e - exists in the resulting list?

Python 2, 59 56 bytes

a,n,l=input()
while l[0]<a:l=[sum(l[:n])]+l
print a in l

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Takes input as A,N,[X,Y]