A171946 N-positions for game of UpMark.

0, 2, 4, 5, 6, 8, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98

Offset: 1

Author

N. J. A. Sloane, Oct 29 2010

Keywords

• nonn
• easy

Comments

It appears that this is the sequence of positions of 0 in the 1-limiting word of the morphism 0 -> 10, 1 -> 00; see A284948. - Clark Kimberling, Apr 18 2017

It appears that this sequence gives the positions of 1 in the limiting 0-word of the morphism 0->11, 1-> 01. See A285383. - Clark Kimberling, Apr 26 2017

Apparently a(n) = 1+A003159(n-1). - R. J. Mathar, Jun 24 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Crossrefs

Complement of A171947.

Programs

Haskell

import Data.List (delete)
a171946 n = a171946_list !! (n-1)
a171946_list = 0 : f [2..] where
   f (w:ws) = w : f (delete (2 * w - 1) ws)
-- Reinhard Zumkeller, Oct 26 2014

Python

def A171946(n):
    if n == 1: return 0
    def f(x):
        c, s = n, bin(x-1)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c += int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 29 2025