A105500 -id:A105500 - cp's OEIS frontend

A278219 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A243353(n)).

1, 2, 4, 2, 4, 8, 6, 2, 4, 12, 16, 8, 6, 12, 6, 2, 4, 12, 36, 12, 16, 32, 24, 8, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 72, 24, 8, 6, 30, 60, 30, 24, 48, 60, 12, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 36, 180, 144, 72, 60, 180, 60, 12, 16, 48, 144, 48, 64, 128, 96, 32, 24, 120, 216, 72, 24, 72

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003188, A046523, A075157, A243353, A278220.
Other base-2 related filter sequences: A278217, A278222.
Sequences that (seem to) partition N into same or coarser equivalence classes are at least these: A005811, A136004, A033264, A037800, A069010, A087116, A090079 and many others like A105500, A106826, A166242, A246960, A277561, A037834, A225081 although these have not been fully checked yet.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]];
Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 93}] (* Michael De Vlieger, May 09 2017 *)

from sympy import prime, factorint
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a(n): return a046523(a243353(n)) # Indranil Ghosh, May 07 2017

(define (A278219 n) (A046523 (A243353 n)))

a(n) = A046523(A243353(n)).
a(n) = A278222(A003188(n)).
a(n) = A278220(1+A075157(n)).

A246960 Directions of the lines in the (Heighway) Dragon Curve.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 0, 3, 2, 3, 2, 1, 2, 3, 0, 3, 0, 1, 0, 3, 2, 3, 0, 3, 2, 3, 2, 1, 2, 3, 0, 3, 0, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 3, 2, 3, 0, 3, 0, 1, 0, 3, 2, 3, 0, 3, 2, 3, 2, 1, 2, 3, 0, 3, 0, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 3, 0, 1, 2, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 3, 2, 3, 0, 3, 0, 1, 0, 3, 0

Offset: 0

Robert G. Wilson v, Sep 08 2014

Fixed point of the morphism: 0 --> 01, 1 --> 21, 2 --> 23, 3 --> 03.

Joerg Arndt, Table of n, a(n) for n = 0..16383
Joerg Arndt, Matters Computational (The Fxtbook), pp. 88-92; image of the dragon curve on p. 89 (top): 0 for north, 1 for west, 2 for south, and 3 for east.

Cf. A014577, A106665.

n where a(n) = 0,1,2,3 respectively: A043724, A043725, A043726, A043727.

Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {2, 3}, 3 -> {0, 3}}] &, {0}, 7]

def A246960(n): return (n^(n>>1)).bit_count()&3 # Chai Wah Wu, Jul 13 2024

a(n) = A005811(n) mod 4. - Joerg Arndt, Sep 09 2014

a(n) = A105500(n) - 1. - Filip Zaludek, Dec 16 2016

A343992 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.

Original entry on oeis.org

0, 1, 0, 1, 3, 0, 0, 6, 3, 20, 0, 0, 29, 0, 0, 56, 101, 108, 0, 392

Offset: 1

N. J. A. Sloane, May 06 2021

Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's starting with D, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that sigma(a) = ab...). Here the letters a,b,c, and d correspond to the four possible steps of the walk. A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism. Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj. Perfect means that four 90-degree rotated copies of the curves Cj started at the origin will pass exactly twice through all grid-points as j tends to infinity (except the origin itself).

It is a theorem that a(A022544(n)) = 0, and a(A001481(n)) > 0 for n>2.

			For n=2 one obtains Heighway's dragon curve, with folding morphism sigma: a -> ab, b -> cb, c -> cd, d -> ad (see A105500 or A246960).

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also arXiv:1011.5788 [math.CO], 2010-2011.

Cf. A296148, A343990, A343991.

Renamed and rewritten by Michel Dekking, Jun 03 2021

A106826 Trajectory of 1 under the morphism 1->{2,1}, 2->{2,3}, 3->{4,3}, 4->{4,1}.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 20 2005

This is the reverse of the morphism in A105500 and the trajectory of 1 actually starts with 2 instead of 1.

F. M. Dekking, Recurrent sets, Advances in Mathematics, vol. 44, no. 1 (1982), 78-104.
Index entries for sequences that are fixed points of mappings

Cf. A105500.

Nest[ Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 3}, 3 -> {4, 3}, 4 -> {4, 1}}] &, {1}, 8] (*Robert G. Wilson v, Jun 20 2005 *)

{a(n)=local(A);if(n<0, 0, n++; A=[2]; while(length(A)

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A278219 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A243353(n)).

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A246960 Directions of the lines in the (Heighway) Dragon Curve.

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A343992 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are perfect.

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A106826 Trajectory of 1 under the morphism 1->{2,1}, 2->{2,3}, 3->{4,3}, 4->{4,1}.

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