A174868 Partial sums of Stern's diatomic series A002487.

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset

0,3

Comments

After the initial 0, identical to A007729.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.

Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Formula

a(n) = Sum_{i=0..n} A002487(i).

G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017

a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018

a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

Example

a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Mathematica

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

Prog

(Python)
from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x, y:(x[0], x[0]+x[1]) if int(y) else (x[0]+x[1], x[1]), bin(n)[-1:2:-1], (1, 0))) for n in count(1)), initial=0)
A174868_list = list(islice(A174868_gen(), 30)) # Chai Wah Wu, May 07 2023

Crossrefs

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

Sequence in context: A157007 A173509 A007729 * A268381 A186349 A304360

Adjacent sequences: A174865 A174866 A174867 * A174869 A174870 A174871

Keyword

nonn,easy

Author

Jonathan Vos Post, Dec 01 2010

Status

approved