A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

1, 3, 7, 11, 19, 27, 39, 51, 71, 91, 119, 147, 187, 227, 279, 331, 403, 475, 567, 659, 779, 899, 1047, 1195, 1383, 1571, 1799, 2027, 2307, 2587, 2919, 3251, 3655, 4059, 4535, 5011, 5579, 6147, 6807, 7467, 8247, 9027, 9927, 10827, 11875, 12923, 14119, 15315

Offset: 0

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000123, A018819, A022907, A033485, A102378, A102379.

Programs

Maple

f:= proc(n) option remember; procname(n-1) + procname(floor(n/2)) + 1 end proc;
f(0):= 1:
map(f, [$1..50]); # Robert Israel, May 04 2025

Mathematica

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]] + 1; Table[a[n], {n, 0, 47}]
nmax = 47; CoefficientList[Series[(1/(1 - x)) (-1 + 2 Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]]}]), {x, 0, nmax}], x]

Python

from itertools import islice
from collections import deque
def A346912_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3, 7)
    while True:
        a += b
        yield 4*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A346912_list = list(islice(A346912_gen(),40)) # Chai Wah Wu, Jun 08 2022

Formula

G.f.: (1/(1 - x)) * (-1 + 2 * Product_{k>=0} 1/(1 - x^(2^k))).

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

a(n) = 2 * A000123(n) - 1.

a(n) = 4 * A033485(n) - 1 for n > 0. - Hugo Pfoertner, Aug 12 2021

From Michael Tulskikh, Aug 12 2021: (Start)

2*a(2n) = a(2n-1) + a(2n+1).

a(2n) = a(2n-2) + a(n-1) + a(n) + 2.

a(2n) = 2*(Sum_{i=0..n} a(i)) - a(n) + 2n. (End)