(A047749) just this one too, then I'll stop

[Meeussen Wouter (bkarnd)]

mat[w_Integer]:= Table[If[j==i+1 || i>j,Mod[i+j,2],0],{i,w},{j,w}] 

mat[w] is defined as :
[ 0,	1,	0,	0,	0,  ... 	]
[ 1,	0,	1,	0,	0 	]
[ 0,	1,	0,	1,	0 	]
[ 1,	0,	1,	0,	1 	]
[ 0,	1,	0,	1,	0 	]
[ 1,	0,	1,	0,	1 	]
[  ...					]

is 1 just above the diagonal,
and below it iff row+column is odd,
else 0.

or substitution game 1->2;2->13;3->24;4->135;
		k_odd->24..(k+1) & k_ev ->13..(k+1)
starting on 1; find length after  n substitutions.

on this matrix, do  I+x A+x^2 A^2 + x^3 A^3 + .. or

GF[w_]:={Prepend[0*Range[w-1],1]}.Inverse[IdentityMatrix[w]-x*mat[w]].Table[{1},{w}]]
or
[[1,0,0,..]] . ( I - x mat)^-1  .  [[1,],[1], ... ]
to get generating functions dependent on size "w".

These produce sequences that approach A047749 at large w.
A parametric GF(w)  can be given in terms of 

f[w_]:= Det[IdentityMatrix[w]-x*mat[w]]

GF[w] is then (x f[w -3] +f[w -2] ) /f[w]    (* Q:  where's the system in this? *)

W.




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