A081706 Numbers n such that binary representation ends either in an odd number of ones followed by one zero or in an even number of ones.

2, 3, 10, 11, 14, 15, 18, 19, 26, 27, 34, 35, 42, 43, 46, 47, 50, 51, 58, 59, 62, 63, 66, 67, 74, 75, 78, 79, 82, 83, 90, 91, 98, 99, 106, 107, 110, 111, 114, 115, 122, 123, 130, 131, 138, 139, 142, 143, 146, 147, 154, 155, 162, 163, 170, 171, 174, 175, 178, 179, 186

Offset: 1

Author

Emeric Deutsch and Bruce E. Sagan, Apr 02 2003

Keywords

• nonn
• base
• easy

Comments

Values of k such that the Motzkin number A001006(k) is even. Values of k such that the number of restricted hexagonal polyominoes with k+1 cells (A002212) is even.

Or union of sequences {2*A079523(n)+k}, k=0,1. A generalization see in comment to A161639. - Vladimir Shevelev, Jun 15 2009

Or intersection of sequences A121539 and {A121539(n)-1}. A generalization see in comment to A161890. - Vladimir Shevelev, Jul 03 2009

Also numbers n for which A010060(n+2) = A010060(n). - Vladimir Shevelev, Jul 06 2009

The asymptotic density of this sequence is 1/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021

Numbers of the form 4^k*(2*n-1)-2 and 4^k*(2*n-1)-1 where n and k are positive integers. - Michael Somos, Oct 22 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
• Jean-Paul Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 2 (2015), pp. 375-388; arXiv preprint, arXiv:1401.3727 [math.NT], 2014.
• Jean-Paul Allouche, André Arnold, Jean Berstel, Srećko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., Vol. 139, No. 1-3 (1995), pp. 455-461.
• Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
• Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A001006, A002212, A003159, A010060, A079523, A121539, A161639, A161890.

Programs

Mathematica

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[200], Mod[m[#], 2] == 0 &] (* Jean-François Alcover, Jul 10 2013 *)
Select[Range[200], EvenQ@Hypergeometric2F1[3/2, -#, 3, 4]&] (* Vladimir Reshetnikov, Nov 02 2015 *)

PARI

is(n)=valuation(bitor(n,1)+1,2)%2==0 \\ Charles R Greathouse IV, Mar 07 2013

Python

from itertools import count, islice
def A081706_gen(): # generator of terms
    for n in count(0):
        if (n&-n).bit_length()&1:
            m = n<<2
            yield m-2
            yield m-1
A081706_list = list(islice(A081706_gen(),30)) # Chai Wah Wu, Jan 09 2023

Python

def A081706(n):
    def f(x):
        c, s = (n+1>>1)+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n+1>>1, f(n+1>>1)
    while m != k: m, k = k, f(k)
    return (m<<2)-1-(n&1) # Chai Wah Wu, Jan 29 2025

Formula

a(2n-1) = 2*A079523(n) = 4*A003159(n)-2; a(2n) = 4*A003159(n)-1.

Note that a(2n) = 1+a(2n-1).