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Go back or double (Posted on 2016-07-25) Difficulty: 2 of 5
1,2,4,3,6,5,10,7,14,8,16,9,18,11,22,12,24...

This sequence has simple rules with a major hint in the title.

Part 1) Determine the rule to extend the sequence.

Part 2) Find, with no aids beyond a simple calculator, the 1000th term of this sequence.

No Solution Yet Submitted by Jer    
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re(2): An uncertain shorthand for part II Comment 5 of 5 |
(In reply to re: An uncertain shorthand for part II by Charlie)

Thank you Charlie for the check


I was thinking doing it on my own, so I try to give form to the relation between the number of term (M)  and the last single number (not doubled) of the sequence (n).

It should be like this: 

M = n + d   [1] being:
  • d=[number of odds + number of multiples of strict even powers of 2] in the interval (n/2,n-1). 
  • "strict even powers of 2" (like 4, 16, 64,...) = it is only multiple of that even power of 2,  (f. ex: is a multiple of 4 but not of 8, 16, etc.).
This could be reduce with good estimation to: 
M= 1 + n + n/4 + n/16 + n/64 + ...
(this work perfect for the numbers 3*2^(2b), as 3, 12, 48, ... 768

To check that for M=1000, n=751 with the above formula [1]: 

For n=751 interval is (376,750).
The number of odds in the interval is (750-376)/2=187

To calculate the number of multiples of strict powers of 2^(2b) in the interval (376-750) a good idea is operate in binary

750=1011101110 [strict multiples of 4= (1011101)+1=94; sm of 16=(10111)=23; sm of 64=(101)+1=6; sm of 256=(1)=1]

376=101111000 (sm of 4=(101111)=47; sm of 16=(1011)+1=12; sm of 64=(10)+1=3]

Then d=187 + (94-47) + (23-12) + (6-3) + 1 = 249

M= n + d = 751+249=1000

Edited on July 27, 2016, 2:04 pm
  Posted by armando on 2016-07-26 16:26:50

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