[seqfan] A165783 Formula for period of shift register stages

[Richard Mathar]

Sequence http://research.att.com/~njas/seuqences/A165783
shows the period length of a shift register which runs
through the map s(k)=s(k-1)+2n-1 if s(k-1) is odd, else s(k)=s(k-1)/2.
(This is obscured in the current OEIS description as
the variable n is used for two different purposes..)
This can be used to extend the sequence to 

2, 3, 6, 4, 9, 15, 18, 5, 12, 27, 8, 15, 30, 27, 42, 6, 15, 17, 54, 16, 30, 21,
    17, 32, 31, 10, 78, 28, 27, 87, 90, 7, 18, 99, 33, 49, 12, 29, 45, 56, 81,
    123, 10, 39, 15, 16, 13, 50, 72, 45, 150, 74, 16, 159, 54, 50, 42, 63, 15,
    33, 165, 26, 150, 8, 21, 195, 26, 53, 102, 207, 69, 85, 42, 62, 222, 20,
    34, 28, 78, 73, 48, 243, 27, 119, 234, 27, 258, 89, 87, 267

I cannot get this to match any formula that one might
derive from correction of parenthesis in the definition line.
Note that a(8)=5<8, so a(n)=n+something_digitsum with a positive semi-definit
digit sum is bothering.

Can anyone with some knowledge on these register sequences
come up with a correction of the formula in the definition, or
is this beyond repair?
I've not received feedback from the author of the sequence.

Richard
# this Maple V9 produces what's shown above:
A165783 := proc(n)
	b := [1] ;
	while true do
		l := nops(b) ;
		if type(op(-1,b),'odd') then
			nextb := op(-1,b)+2*n-1 ;
		else
			nextb := op(-1,b)/2 ;
		fi;
		if member(nextb,b,'w') then
			return nops(b)-w+1 ;
		else
			b := [op(b),nextb] ;
		fi;
	end do;
end:
seq(A165783(n),n=1..90) ;