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G.f. A(x) satisfies: A(x) = 1 + x*A(x^2)/(1 - x)^2.
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%I #10 May 03 2019 11:52:41

%S 1,1,2,4,6,10,14,22,30,44,58,82,106,144,182,242,302,392,482,616,750,

%T 942,1134,1408,1682,2062,2442,2966,3490,4196,4902,5850,6798,8048,9298,

%U 10940,12582,14706,16830,19570,22310,25800,29290,33722,38154,43720,49286,56260,63234,71890,80546

%N G.f. A(x) satisfies: A(x) = 1 + x*A(x^2)/(1 - x)^2.

%H Robert Israel, <a href="/A307889/b307889.txt">Table of n, a(n) for n = 0..10000</a>

%p N:=100: # to get a(1)..a(N)

%p A:= 1:

%p for iter from 1 do

%p B:= convert(series(1 + x*subs(x=x^2,A)/(1-x)^2, x, N+1),polynom);

%p if B = A then break fi;

%p A:= B;

%p od:

%p seq(coeff(A,x,j),j=0..N); # _Robert Israel_, May 03 2019

%t terms = 50; A[_] = 0; Do[A[x_] = 1 + x A[x^2]/(1 - x)^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

%t a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + a[Floor[n/2]]; Join[{1, 1}, Differences[Table[2 a[n + 1], {n, 50}]]]

%Y Cf. A001906, A040039, A298414.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 03 2019