A189663 Partial sums of A189661.

0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31

Offset: 1

Author

Clark Kimberling, Apr 25 2011

Keywords

• nonn

Comments

See A189661.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A001622, A189661, A189662, A026356, A060144.

Programs

Magma

[2*(n-1)-Floor((n-1)*(1+Sqrt(5))/2): n in [1..100]]; // Vincenzo Librandi, Jun 26 2017

Mathematica

(See A189661.)
Table[2 (n - 1) - Floor[(n - 1) (1 + Sqrt[5]) / 2], {n, 100}] (* Vincenzo Librandi, Jun 26 2017 *)

Python

l=[0, 0, 1]
for n in range(3, 101):
    l.append(n - l[n - 1] - l[n - l[n - 1]])
print(l[1:]) # Indranil Ghosh, Jun 24 2017, after Altug Alkan

Python

from math import isqrt
def A189663(n): return (n-1<<1)-(n-1+isqrt(5*(n-1)**2)>>1) # Chai Wah Wu, Aug 09 2022

Formula

a(n) = 2*(n-1)-floor((n-1)*r), where r = (1+sqrt(5))/2 (the golden ratio). - corrected by R. J. Mathar, Sep 11 2011

a(n) = a(1+a(n-2))+a(n-1-a(n-2)), n>2. - Frank Ruskey, Dec 10 2011

a(1) = 0, a(2) = 1; a(n) = n - a(n-1) - a(n-a(n-1)) for n > 2. - Altug Alkan, Jun 24 2017

a(n) = ceiling((n-1)/r^2), where r = (1+sqrt(5))/2. - Jeffrey Shallit, Jul 02 2018

a(n) = A060144(n-1) + sign(abs(n-1)). - Primoz Pirnat, Dec 29 2020