A058039 -id:A058039 - cp's OEIS frontend

A067868 a(n) = a(n-1) + a(floor(n/2))^2 for n > 0, a(0) = 1.

1, 2, 6, 10, 46, 82, 182, 282, 2398, 4514, 11238, 17962, 51086, 84210, 163734, 243258, 5993662, 11744066, 32120262, 52496458, 178789102, 305081746, 627715190, 950348634, 3560128030, 6169907426, 13261231526, 20352555626, 47161378382, 73970201138, 133144655702, 192319110266

Offset: 0

Reinhard Zumkeller, Feb 16 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000123, A056520, A058039.

[1] cat [n le 2 select Factorial(n+1) else Self(n-1) + (Self(Floor(n/2)))^2: n in [1..51]]; // G. C. Greubel, Feb 10 2021

a[n_]:= a[n] = If[n==0, 1, a[n-1] + (a[Floor[n/2]])^2];
Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 10 2021 *)

PARI
```
a(n) = if (n==0, 1, a(n-1)+a(n\2)^2);
```

def a(n): return 1 if n == 0 else a(n-1) + (a(n//2))^2
[a(n) for n in range(50)] # G. C. Greubel, Feb 10 2021

Corrected name and more terms from Michel Marcus, Feb 05 2021

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

Original entry on oeis.org

1, 3, 5, 11, 17, 27, 37, 59, 81, 115, 149, 203, 257, 331, 405, 523, 641, 803, 965, 1195, 1425, 1723, 2021, 2427, 2833, 3347, 3861, 4523, 5185, 5995, 6805, 7851, 8897, 10179, 11461, 13067, 14673, 16603, 18533, 20923, 23313, 26163, 29013, 32459, 35905, 39947, 43989

Offset: 1

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A033485, A033489, A058039.

Partial sums of A039722.

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

from collections import deque
from itertools import islice
def A347027_gen(): # generator of terms
    aqueue, f, b, a = deque([3]), True, 1, 3
    yield from (1, 3)
    while True:
        a += 2*b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A347027_list = list(islice(A347027_gen(),40)) # Chai Wah Wu, Jun 08 2022

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

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

cp's OEIS Frontend

A067868 a(n) = a(n-1) + a(floor(n/2))^2 for n > 0, a(0) = 1.

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