A007302 -id:A007302 - cp's OEIS frontend

A216195 Abelian complexity function of the period-doubling sequence (A096268).

2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 4, 3, 3, 2, 3, 3, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 3, 3, 2, 3, 3, 4, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 3, 4, 4, 4, 3, 4, 3, 3, 2, 3, 3, 4, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 4, 4, 3, 4, 4, 5

Offset: 1

Nathan Fox, Mar 12 2013

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Nathan Fox, Python code to generate sequence

Cf. A007302, A096268.

a[n_]:=Count[BitXor[b1=IntegerDigits[n, 2]; b3=IntegerDigits[3*n, 2]; PadLeft[b1, Length[b3]], b3], 1]; Table[a[n] + 1, {n, 1, 100}] (* Vincenzo Librandi, Jan 13 2017 *)

a(1) = 2; a(2n) = a(n); a(4n-1) = a(n) + 1; a(4n+1) = a(n) + 1.

a(n) = A007302(n) + 1.

A216199 Abelian complexity function of the ternary Thue-Morse word (A036577).

Original entry on oeis.org

3, 3, 5, 3, 4, 5, 4, 3, 5, 4, 6, 5, 6, 4, 5, 3, 4, 5, 6, 4, 6, 6, 6, 5, 6, 6, 6, 4, 6, 5, 4, 3, 5, 4, 6, 5, 6, 6, 6, 4, 6, 6, 8, 6, 7, 6, 6, 5, 6, 6, 7, 6, 8, 6, 6, 4, 6, 6, 6, 5, 6, 4, 5, 3, 4, 5, 6, 4, 6, 6, 6, 5, 6, 6, 7, 6, 8, 6, 6, 4, 6, 6, 8, 6, 7, 8, 7, 6, 8, 7, 8, 6, 7, 6, 6, 5, 6, 6, 7

Offset: 1

Nathan Fox, Mar 12 2013

abs(a(n) - (3/2)*(A007302(n) + 1)) <= 1/2.

Nathan Fox, Python code to generate sequence

Cf. A036577, A007302.

A309709 Number of binary digits that change when n is multiplied by 4.

Original entry on oeis.org

0, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 2, 4, 4, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 2, 4, 2, 4, 4, 4, 4, 4, 4, 6, 6, 8, 4, 4, 6, 6, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 4, 6, 4, 6, 6, 6, 6, 6, 2, 4, 4, 6, 2, 2, 4, 4

Offset: 0

Ali Sada, Aug 14 2019

All terms are even.

			00101_2 * 100_2 = 10100_2: 2 bits changed, so a(5) = 2.

David A. Corneth, Table of n, a(n) for n = 0..9999
Joerg Arndt, Matters Computational (The Fxtbook)
Index entries for sequences related to binary expansion of n

Cf. A000120, A048725, A112627.

Cf. also A007302, A069010.

a:= n-> add(i, i=Bits[Split](Bits[Xor](n*4,n))):
seq(a(n), n=0..120);  # Alois P. Heinz, Aug 23 2019

a[n_] := Total@ IntegerDigits[BitXor[n, 4 n], 2]; Array[a, 88, 0] (* Giovanni Resta, Sep 19 2019 *)

A309709(n) = hammingweight(bitxor(n, n<<2)); \\ Antti Karttunen, Aug 22 2019

def A309709(n):
    s = ""
    while n > 0:
        s, n = str(n%2)+s,n//2
    s, s4, i, j = "00"+s, s+"00", 0, 0
    while i < len(s):
        if s[i] != s4[i]:
            j = j+1
        i = i+1
    return j # A.H.M. Smeets, Aug 23 2019

a(n) = A000120(A048725(n)). - Antti Karttunen, Aug 22 2019

a(A112627(n)) = 2*n and A112627(n) is the first position where 2*n occurs in this sequence. - David A. Corneth, Sep 19 2019

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A216195 Abelian complexity function of the period-doubling sequence (A096268).

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A216199 Abelian complexity function of the ternary Thue-Morse word (A036577).

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A309709 Number of binary digits that change when n is multiplied by 4.

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