A081706 -id:A081706 - cp's OEIS frontend

A344346 Numbers k which have an odd number of trailing zeros in their binary reflected Gray code A014550(k).

3, 4, 11, 12, 15, 16, 19, 20, 27, 28, 35, 36, 43, 44, 47, 48, 51, 52, 59, 60, 63, 64, 67, 68, 75, 76, 79, 80, 83, 84, 91, 92, 99, 100, 107, 108, 111, 112, 115, 116, 123, 124, 131, 132, 139, 140, 143, 144, 147, 148, 155, 156, 163, 164, 171, 172, 175, 176, 179, 180

Offset: 1

Amiram Eldar, May 15 2021

Numbers k such that A050605(k-1) is odd.
Numbers k such that A136480(k) is even.
The asymptotic density of this sequence is 1/3.

			3 is a term since its Gray code, 10, has 1 trailing zero, and 1 is odd.
15 is a term since its Gray code, 1000, has 3 trailing zeros, and 3 is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gray Code.
Wikipedia, Gray code.
Index entries for 2-automatic sequences.

Cf. A014550, A050605, A081706, A136480.

Similar sequences: A001950 (Zeckendorf), A036554 (binary), A145204 (ternary), A217319 (quaternary), A232745 (factorial), A342050 (primorial).

Select[Range[180], OddQ @ IntegerExponent[# * (# + 1)/2, 2] &]

def A344346(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, f(n)
    while m != k: m, k = k, f(k)
    return (m<<2)-(n&1) # Chai Wah Wu, Jan 29 2025

a(n) = A081706(n) + 1. - Hugo Pfoertner, May 16 2021

A162647 Numerators associated with denominators A000215(n) approximating the complementary Thue-Morse constant.

Original entry on oeis.org

2, 3, 10, 151, 38506, 2523490711, 10838310072981296746, 199931532107794273605284333428918544791, 68033174967769840440887906939858451149105560803546820641877549596291376780906

Offset: 0

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

If in the sequence of numbers N for which A010060(N+2^n)=1-A010060(N) the odious (evil) terms are
replaced by 1's (0's), we obtain a 2^(n+1)-periodic binary sequence. These are the post-period
binary (base-2) digits of the complementary Thue-Morse constant 1-A014571 = 0.58754596635989240221663...,
which has a continued fraction and convergents 3/5, 7/12, 10/17, 47/80, 151/257, 801/1365,...
The a(n) are numerators of the convergents selected with denominators taken from A000215.

V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880

Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A162634.

a(n)=A000215(n)-A162634(n). For n>=1, a(n+1)=1+(2^(2^n)-1)*a(n) = 1+A051179(n)*a(n).

Edited by R. J. Mathar, Sep 23 2009

A386987 For n >= 2, a(n) is the least r >= 1 such that T(n - r) + ... + T(n - 1) = T(n + 1) + ... + T(n + r) where T(i) is A010060(i).

Original entry on oeis.org

2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3

Offset: 2

Ctibor O. Zizka, Aug 12 2025

nonn

a(n) is from {1, 2, 3, 4}.

			For n = 6: T(6 - r) + ... + T(5) = T(7) + ... + T(6 + r) is true for the least r = 4  because A010060(2) + A010060(3) + A010060(4) + A010060(5) = A010060(7) + A010060(8) + A010060(9) + A010060(10), thus a(6) = 4.

Cf. A010060, A056196, A079523, A081706, A131323, A225822, A249034.

a[n_] := Module[{s = 0, r = 1}, While[r <= n && (r == 1 || s != 0), s += (ThueMorse[n - r] - ThueMorse[n + r]); r++]; r-1]; Array[a, 100, 2] (* Amiram Eldar, Aug 12 2025 *)

a(A081706(n) + 1) = 1.
a(2*A079523(n)) = 2.
a(A249034(n))= 2.
a(A225822(n)) = 3.
a(A056196(n)) = 3.
a(2*A131323(n)) = 4.
a(2*A249034(n) - 1) = 4.

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A344346 Numbers k which have an odd number of trailing zeros in their binary reflected Gray code A014550(k).

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A162647 Numerators associated with denominators A000215(n) approximating the complementary Thue-Morse constant.

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A386987 For n >= 2, a(n) is the least r >= 1 such that T(n - r) + ... + T(n - 1) = T(n + 1) + ... + T(n + r) where T(i) is A010060(i).

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