A346912 - OEIS

%I #42 Jun 08 2022 16:22:32

%S 1,3,7,11,19,27,39,51,71,91,119,147,187,227,279,331,403,475,567,659,

%T 779,899,1047,1195,1383,1571,1799,2027,2307,2587,2919,3251,3655,4059,

%U 4535,5011,5579,6147,6807,7467,8247,9027,9927,10827,11875,12923,14119,15315

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

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

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

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

%F a(n) = 4 * A033485(n) - 1 for n > 0. - _Hugo Pfoertner_, Aug 12 2021

%F From _Michael Tulskikh_, Aug 12 2021: (Start)

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

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

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

%t a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]] + 1; Table[a[n], {n, 0, 47}]

%t 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]

%o (Python)

%o from itertools import islice

%o from collections import deque

%o def A346912_gen(): # generator of terms

%o     aqueue, f, b, a = deque([2]), True, 1, 2

%o     yield from (1, 3, 7)

%o     while True:

%o         a += b

%o         yield 4*a - 1

%o         aqueue.append(a)

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

%o         f = not f

%o A346912_list = list(islice(A346912_gen(),40)) # _Chai Wah Wu_, Jun 08 2022

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Aug 11 2021