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

Author

Jonathan Vos Post, Dec 01 2010

Keywords

• nonn
• easy

Comments

After the initial 0, identical to A007729.

Examples

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

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.

Crossrefs

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

Cf. A001622, A242208.

Programs

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 *)

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

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