A112658 -id:A112658 - cp's OEIS frontend

A099545 Odd part of n, modulo 4.

1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3

Offset: 1

Ralf Stephan, Oct 23 2004

The terms of this sequence are the even-indexed terms of A112658. - Alexandre Wajnberg, Jan 02 2006
Fractal sequence: odd terms are 1, 3, 1, 3,...; the even terms are the sequence itself: a(n)=a(2n)=a(4n)=a(8n)=a(16n)=... - Alexandre Wajnberg, Jan 02 2006
From Micah D. Tillman, Jan 29 2021: (Start)
Has the same structure as the regular paper-folding (dragon curve) sequence (A014577, A014709). We can interpret a(n) as the number of 90-degree rotations to make in a single direction at the n-th "turn" in the dragon curve. After all, making three 90-degree rotations to the left (turning a total of 270 degrees) is equivalent to making one 90-degree rotation to the right, and vice versa.
We can likewise produce the dragon curve by interpreting A000265(n), the whole odd part of n, as the number of 90-degree rotations to make in a single direction at the n-th "turn" in the curve. (End)

			a(100) = 1: the odd part of 100 is 100/4 = 25, and 25 mod 4 = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A010873, A000265, A014707, A038189.

Cf. A099544, A099546, A099547, A099548, A099549, A099550, A099551.

Array[Mod[#/(2^IntegerExponent[#, 2]), 4] &, 105] (* Michael De Vlieger, Feb 24 2021 *)

a(n)=bitand(n/(2^valuation(n,2)), 3); /* Joerg Arndt, Jul 18 2012 */

def A099545(n): return n>>(~n&n-1).bit_length()&3 # Chai Wah Wu, Feb 26 2025

a(n) = 2 * A038189(n) + 1.
(a(n)-1)/2 = A014707(n). - Alexandre Wajnberg, Jan 02 2006
a(n) = A010873(A000265(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Aug 29 2024

A122002 a(0)=5; otherwise a(n) = (n mod 4) if n is odd, a(n) = h + 4, where h = (highest odd divisor of n) mod 4 if n is even.

Original entry on oeis.org

5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 7, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 5, 1, 5, 3, 7, 1, 7, 3, 5, 1, 5, 3, 5, 1, 7, 3, 7, 1, 5, 3, 7, 1, 7, 3, 7, 1, 5, 3, 5

Offset: 0

N. J. A. Sloane, Aug 05 2008

nonn

a(n) in {1,3,5,7} for all n. a(4k+i) = i if i is odd.

There is a typo in Grytczuk's definition: he has "+ 5" instead of "+ 4".

A. Carpi, Multidimensional unrepetitive configurations, Theoret. Comput. Sci., 56 (1988), 233-241. a(n) = a_n of lemma 3.2 for the case p=2 and m=0 (which is corollary 3.3).
Jaroslaw Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429. [See Problem 15.]
Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, Manuel Silva, Words Avoiding Repetitions in Arithmetic Progressions, Theoretical Computer Science, volume 391, issues 1-2, February 2008, pages 126-137. And arXiv:math/0608607 [math.CO], 2006. (Extending to generalized paperfolding sequences.)
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to squarefree words

Essentially the same: A112658 (map 1357 -> 0213), A125047 (map 1357 -> 2314).

Cf. A003324.

a[0]=5;a[n_]:=If[OddQ[n],Mod[n,4],4+Mod[Select[Divisors[n],OddQ][[-1]],4]];Table[a[n],{n,0,100}] (* James C. McMahon, Oct 25 2024 *)

a(n) = 2*if(n,bittest(n,valuation(n,2)+1)) + if(n%2,1,5); \\ Kevin Ryde, Sep 09 2020

Morphism 1 -> 5,3; 3 -> 7,3; 5 -> 5,1; 7 -> 7,1 starting from 5 [Carpi, h in remark after lemma 3.2]. - Kevin Ryde, Sep 09 2020

Terms from a(26) on from John W. Layman, Aug 27 2008

A125047 Infinite word generated by mapping 1->12, 2->13, 3->43, 4->42 starting at 1.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 17 2006

nonn

Infinite word over 4-letter alphabet that contains no squares in arithmetic progressions of odd difference. - Ralf Stephan, May 09 2007

			1 -> 12 -> 1213 -> 12131242 -> 1213124312134243 -> ...

Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, Manuel Silva, Words avoiding repetitions in arithmetic progressions, Theoretical Computer Science, volume 391, issues 1-2, February 2008, pages 126-137. And arXiv:math/0608607 [math.CO], 2006.
Index entries for sequences that are fixed points of mappings

Essentially the same: A112658 (map 1234 -> 1023), A122002 (map 1234 -> 5137).

Cf. A038190.

SubstitutionSystem[{1 -> {1, 2}, 2 -> {1, 3}, 3 -> {4, 3}, 4 -> {4, 2}}, {1}, 7] // Last (* Jean-François Alcover, Dec 17 2018 *)

{a(n)=local(A); if(n<1, 0, A=[1]; while(length(A)

my(table=[1,2;4,3]); a(n) = n--; table[if(n,bittest(n,1+valuation(n,2)))+1, n%2+1]; \\ Kevin Ryde, Sep 05 2020

Recurrence: a(1)=1, a(4n)=3, a(4n+2)=2, a(8n+3)=1, a(8n+7)=4, a(4n+1)=a(2n+1). - Ralf Stephan, May 09 2007

A343421 Number of Dean words of length n, i.e., squarefree reduced words over {0,1,2,3}.

Original entry on oeis.org

4, 8, 16, 24, 40, 64, 104, 144, 216, 328, 496, 720, 1072, 1584, 2344, 3384, 4952, 7264, 10632, 15504, 22656, 33136, 48488, 70592, 103032, 150352, 219400, 319816, 466664, 680872, 993440, 1447952, 2111448, 3079464, 4491216, 6548936, 9550728, 13927840, 20311168

Offset: 1

Michel Marcus, Apr 15 2021

nonn

A Dean word is a reduced word that does not contain occurrences of ww for any nonempty w.

a(n) is a multiple of 4 by symmetry. - Michael S. Branicky, Jun 20 2022

Martin Ehrenstein, Table of n, a(n) for n = 1..64
Richard A. Dean, A sequence without repeats on x, x^{-1}, y, y^{-1}, Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.
Tero Harju, Avoiding Square-Free Words on Free Groups, arXiv:2104.06837 [math.CO], 2021.

Cf. A003324, A112658, A051041.

def isf(s): # incrementally squarefree (check factors ending in last letter)
    for l in range(1, len(s)//2 + 1):
        if s[-2*l:-l] == s[-l:]: return False
    return True
def aupton(nn, verbose=False):
    alst, sfs = [], set("0")
    for n in range(1, nn+1):
        an, eo = 4*len(sfs), ["02", "13"]
        sfsnew = set(s+i for s in sfs for i in eo[n%2] if isf(s+i))
        alst, sfs = alst+[an], sfsnew
        if verbose: print(n, an)
    return alst
print(aupton(30)) # Michael S. Branicky, Jun 20 2022

More terms from James Rayman, Apr 15 2021

A343180 Trajectory of 1 under the morphism 1 -> 12, 2 -> 32, 3 -> 14, 4 -> 34.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 02 2021

nonn

J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. See page 6.
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. See page 6. [Local copy]

A112658 is another version of the same sequence.

f(1):= (1,2): f(2):= (3,2): f(3) := (1,4); f(4) := (3,4);  #
A:= [1]:
for i from 1 to 8 do A:= map(f, A) od:
A;

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A099545 Odd part of n, modulo 4.

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A122002 a(0)=5; otherwise a(n) = (n mod 4) if n is odd, a(n) = h + 4, where h = (highest odd divisor of n) mod 4 if n is even.

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A125047 Infinite word generated by mapping 1->12, 2->13, 3->43, 4->42 starting at 1.

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A343421 Number of Dean words of length n, i.e., squarefree reduced words over {0,1,2,3}.

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Python

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A343180 Trajectory of 1 under the morphism 1 -> 12, 2 -> 32, 3 -> 14, 4 -> 34.

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