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A347027 a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).

%I A347027 #11 Jun 24 2022 19:45:25
%S A347027 1,3,5,11,17,27,37,59,81,115,149,203,257,331,405,523,641,803,965,1195,
%T A347027 1425,1723,2021,2427,2833,3347,3861,4523,5185,5995,6805,7851,8897,
%U A347027 10179,11461,13067,14673,16603,18533,20923,23313,26163,29013,32459,35905,39947,43989
%N A347027 a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).
%F A347027 G.f. A(x) satisfies: A(x) = (x + 2 * (1 + x) * A(x^2)) / (1 - x).
%F A347027 a(n) = 1 + 2 * Sum_{k=2..n} a(floor(k/2)).
%t A347027 a[1] = 1; a[n_] := a[n] = a[n - 1] + 2 a[Floor[n/2]]; Table[a[n], {n, 1, 47}]
%t A347027 nmax = 47; A[_] = 0; Do[A[x_] = (x + 2 (1 + x) A[x^2])/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%o A347027 (Python)
%o A347027 from collections import deque
%o A347027 from itertools import islice
%o A347027 def A347027_gen(): # generator of terms
%o A347027     aqueue, f, b, a = deque([3]), True, 1, 3
%o A347027     yield from (1, 3)
%o A347027     while True:
%o A347027         a += 2*b
%o A347027         yield a
%o A347027         aqueue.append(a)
%o A347027         if f: b = aqueue.popleft()
%o A347027         f = not f
%o A347027 A347027_list = list(islice(A347027_gen(),40)) # _Chai Wah Wu_, Jun 08 2022
%Y A347027 Cf. A033485, A033489, A058039.
%Y A347027 Partial sums of A039722.
%K A347027 nonn
%O A347027 1,2
%A A347027 _Ilya Gutkovskiy_, Aug 11 2021