Combining sequences: Take 2n and 2n-1 where n is a natural number...

[murphy1135]

Take 2n and 2n-1 where n is a natural number
The first sequence yields 2,4,6,8,...
The second yields 1,3,5,7,...
Combining them yields 1,2,3,4,... or the natural numbers
or simply the sequence n.
Is there a general way to get the result (in this case the sequence n) without having to go through the sequences?


What I find difficult is...
I have the following sequences:
18n-13,18n-11,18n-1,18n-5,18n-7,18n-17,6n-3
I know that combining these sequences will be the same as the sequence
2n-1 or all the odd numbers. But apart from manually doing it I do not know a general way to get there.


An example of doing it manually:
A. 6n-3 yields 3,9,15,21,27,33,39...
B. 18n-13 yields 5,23,41,59,...
C. 18n-11 yields 7,25,43,61,79,...
D. 18n-1 yields 17,35,53,71,...
E. 18n-5 yields 13,31,49,67,85,...
F. 18n-7 yields 11,29,47,65,83,...
G. 18n-17 yields 1,19,37,55,73,...
Combining all these sequences on a number line yields 1,3,5,7...
or the sequence 2n-1
The sequence of elements are as follows:
G1,A1,B1,C1,A2,F1,E1,A3,D1,
G2,A4,B2,C2,A5,F2,E2,A6,D2,
G3,A7,B3,C3,A8,F3,E3,A9,D3,...


So I know that in the 1,3,5,7 sequence each other sequence behaves as follows in terms of position:
G: 9n-8 ... G is in positions 1,10,19,28....
F: 9n-3 ... F is in positions 6,15,24,33...
E: 9n-2 ... E is in positions 7,16,25,34...
D: 9n ... D is in positions 9,18,27,...
C: 9n-5... C is in positions 4,13,22,31,...
B: 9n-6... B is in positions 3,12,21,30,...
A: 3n-1... A is in positions 2,5,8,11...


This set of sequences gives the natural numbers.
If we take the second G ie.. 9n-8 and we do 2G-1 then we get back to 18n-17 so I can see the relationship there.


The problem still being that for both sets of sequences I cannot see a short way to derive the equivalent sequences (2n-1 and n respectively.)


Is there a shorter numerical way if you know your starting sequences to calculate what the resulting sequence would be?

 

[murphy1135]

Are you just fiddling around, or is this a classroom assignment?

This is purely for my own curiosity.

I know that combining some sequences produce "unrelated" results:
Example: Combining n^2 (squares) and n*n-1 *(oblong numbers) yields Quarter squares ie.. 0,0,1,2,4,6,8,12,16... (which can be constructed by floor(n^2/4).
Now it is apparent that there is no direct way to derive the result formula from the starting ones... You are left with having to recognize the sequence for what it is.

My question above has to do with related sequences. In other words... Is there a general way to evaluate some sets of formulations eg 6n-3,18n-3... to yield the resulting formulation eg 2n-1
Am perfectly happy if there is not.