A051069 -id:A051069 - cp's OEIS frontend

A051065 a(n) = A004128(n) mod 2.

0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane and Gary W. Adamson

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Benoit Cloitre, 3^13 steps of the walk using step: turn +90 degrees if a(n)=0, -90 degrees if a(n)=1.

Cf. A004128, A051066, A051067, A051069.

Join[{0}, Mod[Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]], 2]] (* Amiram Eldar, Jun 02 2025 *)

a(n)=if(n<1,0,(a(n\3)+n)%2) \\ Benoit Cloitre, Nov 21 2013

TOP = 1000
a = [0]*TOP
for n in range(1, TOP):
    print(a[n-1], end=',')
    a[n] = (n + a[n//3]) % 2
# Alex Ratushnyak, Aug 17 2012

def A051065(n):
    c, m, a = 1, 3*n, 0
    for k in range(1,n+1):
        c *= 3
        if c > m:
            break
        a ^= m//c&1
    return a # Chai Wah Wu, Sep 02 2025

a(0)=0, a(n) = (n + a(floor(n/3))) mod 2. - Alex Ratushnyak, Aug 17 2012

More terms from James Sellers, Dec 11 1999

A051068 Partial sums of A014578.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 49

Offset: 0

N. J. A. Sloane, Gary W. Adamson

nonn

Duplicate of A050294? [Joerg Arndt, Apr 27 2013]
From Michel Dekking, Feb 10 2019: (Start)
The answer to Joerg Arndt's question is: yes (modulo an offset).  To see this, it suffices to prove that the two sequences of first differences Da and Db of a= A051068 and b:=A050294 are equal. Clearly the sequence Da of first differences of a is the sequence A014578. According to Philippe Deleham (2004), Da equals 0x = 0110110111110..., where x is the fixed point of the morphism 0->111, 1->110.
From Vladimir Shevelev (2011) we know a formula for b=A050294: b(n) = n-b(floor(n/3)). This gives that the sequence of first differences Db:=(b(n+1)-b(n)) of b satisfies
      Db(3m+1) = Db(3m+2) = 1, and Db(3m+3) = 1 - Db(m).
This implies that Db = x, the fixed point of 0->111, 1->110.
(End)

Cf. A014578, A051069, A050294.

a(3^n) = A015518(n+1) = -(-1)^n*A014983(n+1). - Philippe Deléham, Mar 31 2004

A092400 Fixed point of the morphism 1 -> 1121211, 2 -> 1121212121211, starting from a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Philippe Deléham, Mar 21 2004

Length of n-th run of identical symbols in A051069.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences that are fixed points of mappings.

Cf. A007417, A051069.

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 1, 2, 1, 2, 1, 1}, 2 -> {1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1}})]}], {1}, 3] (* Robert G. Wilson v, Feb 26 2005 *)

from sympy import integer_log
def A007417(n):
    def f(x): return n+x-sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m
def A092400(n): return A007417(n)-A007417(n-1) if n>1 else 1 # Chai Wah Wu, Feb 16 2025

Sum_{k=1..n} a(k) = A007417(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/3. - Amiram Eldar, Apr 11 2025

More terms from Robert G. Wilson v, Feb 26 2005

cp's OEIS Frontend

A051065 a(n) = A004128(n) mod 2.

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A051068 Partial sums of A014578.

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A092400 Fixed point of the morphism 1 -> 1121211, 2 -> 1121212121211, starting from a(1) = 1.

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