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%I #22 Feb 08 2022 23:38:48
%S 1,1,2,3,6,10,18,32,58,103,184,329,588,1051,1878,3357,6000,10723,
%T 19164,34251,61214,109404,195530,349458,624562,1116237,1994974,
%U 3565481,6372340,11388848,20354510,36378224,65016314,116199213,207674912,371163175
%N An INVERT transform of the Thue-Morse sequence.
%C Eigensequence for sequence array of A010060(n+1). - _Paul Barry_, Nov 03 2010
%C Starting with offset 1 represents the numbers of ordered compositions of n using the odious numbers: (1, 2, 4, 7, 8, 11, ...). Cf. A000069. - _Gary W. Adamson_, Apr 04 2013
%F G.f.: 1/(1-(1/2)*(1/(1-x) - (Product_{k>=0} (1-x^(2^k))))). - _Paul Barry_, Nov 03 2010
%F a(n) = Sum_{k=0..n-1} A010060(n-k)*a(k) with a(0) = 1. - _Johannes W. Meijer_, Jun 19 2012
%e a(4) = A010060(4)*a(0) + A010060(3)*a(1) + A010060(2)*a(2) + A010060(1)*a(3) = (1, 0, 1, 1) dot (1, 1, 2, 3) = 1 + 0 + 2 + 3 = 6.
%p A010060:=proc(n) local n2: n2:=convert(n, base, 2): sum(n2[j], j=1..nops(n2)) mod 2; end: a:=proc(n) option remember: if n=0 then 1 else a(n) := add(A010060(n-k)*a(k),k=0..n-1) fi: end: seq(a(n), n=0..34); # _Johannes W. Meijer_, Jun 19 2012
%o (PARI) a(n)=polcoeff(1/(1-sum(i=1,n,(hammingweight(i)%2)*x^i)+O(x^(n+1))),n) \\ _Ralf Stephan_, Dec 10 2013
%Y Cf. A010060, A144027.
%Y Cf. A000069.
%K nonn,easy
%O 0,3
%A _Gary W. Adamson_, Sep 07 2008
%E Edited by _Johannes W. Meijer_, Jun 19 2012
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