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

1,4

Comments

See A189661.

Links

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A057367 A032634 A057366 * A341440 A355028 A061375

Adjacent sequences: A189660 A189661 A189662 * A189664 A189665 A189666

Keyword

nonn

Author

Clark Kimberling, Apr 25 2011

Status

approved