Am I a Cullen Number?

Question

A Cullen Number is any number that is contained in the sequence generated using the formula:

C(n) = (n*2^n)+1.

Your Task:

Write a program or function that receives an input and outputs a truthy/falsy value based on whether the input is a Cullen Number.

Input:

A non-negative integer between 0 and 10^9 (inclusive).

Output:

A truthy/falsy value that indicates whether the input is a Cullen Number.

Test Cases:

Input:    Output:
1   --->  truthy
3   --->  truthy
5   --->  falsy
9   --->  truthy
12  --->  falsy
25  --->  truthy

Scoring:

This is code-golf, so the lowest score in bytes wins.

Answer 1

QBIC, 24 bytes

[0,:|~a*(2^a)+1=b|_Xq}?0

Explanation

[0,:|           FOR a = 0 to b (input from cmd line)
~a*(2^a)+1=b    IF calculating this a results in b
|_Xq            THEN quit, printing 1
}               NEXT a
?0              We haven't quit early, so print 0 and end.

Answer 2

k, 19 bytes

{&1=x-{x**/x#2}'!x}

Try it online. Truthy is an array with a number in it: ,3 or ,0 et cetera. Falsey is an empty array: () or !0 depending on your interpreter.

Answer 3

Java (OpenJDK 8), 56 bytes

i->{int n,c;for(n=0;(c=n*(2<<n++-1)+1)<i;);return c==i;}

Try it online!

Answer 4

Pari/GP, 25 bytes

n->!prod(i=0,n,n-i*2^i-1)

Try it online!

Answer 5

Thunno 2 -c, 5 bytes

ĖıỴ;Ƈ

Try it online!

Port of user62131's Jelly answer.

Thunno 2, 6 bytes

ĖDO×⁺Ƈ

Try it online!

Explanation

ĖıỴ;Ƈ  # Implicit input
       # Decrement the input
Ė      # Push [0..input-1]
  Ỵ    # Left bit-shift each
 ı ;   # number by itself
    Ƈ  # Is (input - 1)
       # in this list?
       # Implicit output

ĖDO×⁺Ƈ  # Implicit input
Ė       # Push [0..input]
 D      # Duplicate it
  O     # Push 2 ** that
   ×    # Multiply together
    ⁺   # Increment each
     Ƈ  # Is the input
        # in this list?
        # Implicit output