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a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).
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%I #11 Jun 24 2022 19:45:25

%S 1,3,5,11,17,27,37,59,81,115,149,203,257,331,405,523,641,803,965,1195,

%T 1425,1723,2021,2427,2833,3347,3861,4523,5185,5995,6805,7851,8897,

%U 10179,11461,13067,14673,16603,18533,20923,23313,26163,29013,32459,35905,39947,43989

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

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

%F a(n) = 1 + 2 * Sum_{k=2..n} a(floor(k/2)).

%t a[1] = 1; a[n_] := a[n] = a[n - 1] + 2 a[Floor[n/2]]; Table[a[n], {n, 1, 47}]

%t 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 (Python)

%o from collections import deque

%o from itertools import islice

%o def A347027_gen(): # generator of terms

%o aqueue, f, b, a = deque([3]), True, 1, 3

%o yield from (1, 3)

%o while True:

%o a += 2*b

%o yield a

%o aqueue.append(a)

%o if f: b = aqueue.popleft()

%o f = not f

%o A347027_list = list(islice(A347027_gen(),40)) # _Chai Wah Wu_, Jun 08 2022

%Y Cf. A033485, A033489, A058039.

%Y Partial sums of A039722.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Aug 11 2021