[seqfan] A003312, a limit of the decomposition

[reismann at free.fr]

Dear seqfans,

I was looking for the sequence with the largest growth (the largest jumps) which is decomposable into weight + level + jump and I found that it is A003312 :
A003312 weight level jump
3 =     2 *    1 +   1
4 =     3 *    1 +   1
5 =     3 *    1 +   2
7 =     4 *    1 +   3
10 =    6 *    1 +   4
14 =    8 *    1 +   6
20 =    11 *   1 +   9
29 =    15 *   1 +   14
43 =    22 *   1 +   21
64 =    33 *   1 +   31
...

I notice that if A003312(n) is even then jump = weight + 2 else  jump = weight + 1.
The sequence of jumps (gap, first difference) seems to be A005428 but there is no crossref beteween these sequences, so I am not sure. Could someone confirm ?

If I am not wrong, I would like to add on the OEIS : 
- a formula to A005428
%F a(n) = A003312(n+1) - A003312(n).

- the sequence of weights of A003312 (AXXXXXX)
%S 2,3,3,4,6,8,11,15,22,33,... (I need a b-file 1..500, any help would be appreciated)
%N a(n) = smallest k such that A003312(n+1) = A003312(n) + (A003312(n) mod k)

- a comment on A003312 : 
%C The decomposition of a(n) into weight * level + jump is : a(n) = AXXXXXX * 1 + A003312
and maybe a comment to say that a(n) is a limit sequence of the decomposition.

Best,

Rémi.