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Defining a sequence (Posted on 2018-10-31) Difficulty: 3 of 5
What binary sequence a1,a2,a3,...an,... can be defined by the equation: (an)XOR (3*an)=an+1?
Provide the proof of your formula.

No Solution Yet Submitted by Ady TZIDON    
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XOR distributive Comment 3 of 3 |
Heavily modified comment: 
Yes, the operation f(a) = (a) XOR (3a) is a doubling. 
f(a)= 2 a. 

Since I proved it for a single bit a_1 = 1, and an arbitrary bit b, a_b = 1, I think I unknowingly proved it for all numbers (!) since all numbers are superpositions of n bits. Since the relationship holds for all single bits in position b starting with a_n = a_b then it holds for all numbers: f(a_i + a_j + a_k ... ) = (2a_i + 2a_j + 2a_k) = 
2 (a_i+a_j + a+k ...)   
 
one thing needed to complete the proof is this: 
for the case:  a_n = 0 
f(a_n) = 0 = 2 a_n

Edited on October 31, 2018, 11:25 am
  Posted by Steven Lord on 2018-10-31 10:34:13

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