[seqfan] A new sequence?

[DAN_CYN_J]

Hi all sequence fans, 

  

A nice new sequence related to A026272. 

  

A sequence where all even positive integers are represented exactly twice. 
Also the even gap count between the integer represent that integer. 

  

2,4,6,2,8,10,4,12,14,6,16,18,20,8,22,24,10,26,28,30,12,32,34,14,36,38,40,16,42,44,18, 
46,48,20,50,52,54,22,56,58,24,... 

It is related to the silver mean whose cf = [2:2,2,2,2,2,2,2,2,2,2,2,...] 
= 
2.4142135623730950488016887242096980785696718753769480731766797379907324784... 

When a(n) value is < a(n-1) then a(n-1)/a(n) will converge too the silver mean. 

  

Just like A026272 --- 1,2,1,3,2,4,5,3,6,7,4,8,5... sequence where certain 
consecutive integers converges too phi where the best phi convergence happens 
at the Fibonacci pairs within the sequence.  [3,2],[5,3],[8,5],[13,8]... 

  

I am not sure other sequences can be produced related to this same format 
to converge on the next larger metal mean such as cf =[3:3,3,3,3,3,3,3...] ? 

  

Dan