A163356 -id:A163356 - cp's OEIS frontend

A163912 Least common multiple of all cycle sizes in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 6, 24, 36, 288, 432, 1728, 2592, 31104, 15552

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163892, A163910, A163911, A163904, A163890.

A163910 Number of cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 3, 18, 30, 178, 306, 1864, 3214, 20032, 34708

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163911, A163912, A163914, A163904, A163890.

A163911 Maximum cycle size in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 6, 8, 18, 32, 108, 216, 324, 1944, 1944

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163891, A163896, A163910, A163912, A163904, A163890.

A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

0, 0, 2, 1, 10, 9, 54, 57, 295, 329, 1613, 1834, 8812, 10072

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = A163913(n)/3. Bisections: A163909, A163919. See also A163903, A163911, A163912, A163904, A163890.

A163906 Permutation A163356 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 12, 15, 13, 14, 7, 4, 5, 6, 9, 11, 8, 10, 16, 19, 17, 18, 20, 21, 22, 23, 27, 25, 26, 24, 30, 29, 28, 31, 47, 44, 45, 46, 39, 37, 38, 36, 40, 41, 42, 43, 34, 33, 35, 32, 58, 56, 59, 57, 54, 53, 52, 55, 49, 50, 48, 51, 60, 61, 62, 63, 192, 195, 193, 194, 196, 197

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Inverse: A163905. a(n) = A163356(A163356(n)). See also A163916.

A163916 Permutation A163356 applied thrice.

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 4, 9, 8, 10, 11, 14, 13, 12, 15, 16, 17, 18, 19, 20, 21, 23, 22, 30, 29, 31, 28, 24, 25, 27, 26, 58, 59, 57, 56, 54, 53, 55, 52, 60, 61, 63, 62, 51, 50, 49, 48, 32, 35, 34, 33, 36, 37, 39, 38, 46, 44, 47, 45, 40, 41, 43, 42, 127, 125, 126, 124, 119, 117

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Inverse: A163915. Cf. A163356, A163906.

a(n) = A163356(A163356(A163356(n))) = A163356(A163906(n)).

A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 15, 14, 13, 10, 9, 8, 11, 4, 5, 6, 7, 24, 27, 26, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 21, 20, 23, 40, 43, 42, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 37, 36, 39, 56, 57, 58, 59, 52, 55, 54, 53, 50, 49, 48, 51, 60, 61, 62, 63, 192, 195, 194, 193, 198, 199, 196, 197, 202, 203, 200, 201, 206, 205

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

When A207901, which is a multiplicative walk permutation, is composed from the right with this permutation, the result is A302781, another multiplicative walk permutation.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302843 (inverse).

Cf. A003188, A006068, A057300, A163356, A302846, A302781.

A003188(n) = bitxor(n, n>>1);
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r)))); \\ Antti Karttunen, Apr 14 2018
A302844(n) = A003188(A163356(n));

a(n) = A003188(A163356(n)).

a(n) = A006068(A302846(n)).

A163913 Number of integers i in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356 with A163915(i)=i, but not A163355(i)=i.

Original entry on oeis.org

0, 0, 6, 3, 30, 27, 162, 171, 885, 987, 4839, 5502, 26436, 30216

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = 3*A163914(n). See also A163903.

A163355 Permutation of integers for constructing Hilbert curve in N x N grid.

Original entry on oeis.org

0, 1, 3, 2, 14, 15, 13, 12, 4, 7, 5, 6, 8, 11, 9, 10, 16, 19, 17, 18, 20, 21, 23, 22, 30, 29, 31, 28, 24, 25, 27, 26, 58, 57, 59, 56, 54, 53, 55, 52, 60, 61, 63, 62, 50, 51, 49, 48, 32, 35, 33, 34, 36, 37, 39, 38, 46, 45, 47, 44, 40, 41, 43, 42, 234, 235, 233, 232, 236, 239

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..262143
Index entries for sequences that are permutations of the natural numbers

Inverse: A163356. A163357 & A163359 give two variants of Hilbert curve in N x N grid. Cf. also A163332.
Second and third "powers": A163905, A163915.
In range [A000302(n-1)..A024036(n)] of this permutation, the number of cycles is given by A163910, number of fixed points seems to be given by A147600(n-1) (fixed points themselves: A163901). Max. cycle sizes is given by A163911 and LCM's of all cycle sizes by A163912.
See also: A000695, A163890, A163894, A163902-A163903, A163914, A163485, A302843, A302845.

A057300 := proc(n)
    option remember;
    `if`(n=0, 0, procname(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1])
end proc:
A163355 := proc(n)
    option remember ;
    local d,base4,i,r ;
    if n <= 1 then
        return n ;
    end if;
    base4 := convert(n,base,4) ;
    d := op(-1,base4) ;
    i := nops(base4)-1 ;
    r := n-d*4^i ;
    if ( d=1 and type(i,even) ) or ( d=2 and type(i,odd)) then
        4^i+procname(A057300(r)) ;
    elif d= 3 then
        2*4^i+procname(A057300(r)) ;
    else
        3*4^i+procname(4^i-1-r) ;
    end if;
end proc:
seq(A163355(n),n=0..100) ; # R. J. Mathar, Nov 22 2023

A057300(n) = { my(t=1, s=0); while(n>0,  if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r)))); \\ Antti Karttunen, Apr 14 2018

a(0) = 0, and given d=1, 2 or 3, then a((d*(4^i))+r)
  = (4^i) + a(A057300(r)), if d=1 and i is even, or if d=2 and i is odd
  = 2*(4^i) + a(A057300(r)), if d=3,
  = 3*(4^i) + a((4^i)-1-r) in other cases.
From Alan Michael Gómez Calderón, May 06 2025: (Start)
a(3*A000695(n)) = 2*A000695(n);
a(3*(A000695(n) + 2^A000695(2*m))) = 2*(A000695(n) + 2^A000695(2*m)) for m >= 2;
a((2 + 16^n)*2^(-1 + 4*m)) = 4^(2*(n + m) - 1) + (11*16^m - 2)/3. (End)

Links to further derived sequences added by Antti Karttunen, Sep 21 2009

A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 5, 5, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 6, 6, 7, 7, 7, 6, 6, 5, 4, 4, 5, 5, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 14, 14, 15, 15, 14

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the X-coordinate of the n-th term in Hilbert's Hamiltonian walk A163359 and the Y-coordinate of its transpose A163357.

			[m(1)=0 0 1 1, m'(1)= 0 1 10] [m(2) =0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, m'(2)=0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3].

Antti Karttunen, Table of n, a(n) for n = 0..65535
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.1 "The Hilbert curve", page 85, lin2hilbert.
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 115 by Gosper, page 52. Also HTML transcription. (To use algorithm S or the state table, pad n with high 0-bits to a multiple of 4 bits.)
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m'(3), A059253, as well as: A163539, A163540, A163542, A059261, A059285, A163547 and A163529.

void h(unsigned int *x, unsigned int *y, unsigned int l){
x[0] = y[0] = 0; unsigned int *t = NULL; unsigned int n = 0, k = 0;
for(unsigned int i = 1; i>(2*n)){
case 1: x[i] = y[i&k]; y[i] = x[i&k]+(1<Jared Rager, Jan 09 2021 */
(C++) See Fxtbook link.

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A002262(A163358(n)) = A025581(A163360(n)) = A059906(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

cp's OEIS Frontend

A163912 Least common multiple of all cycle sizes in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163910 Number of cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163911 Maximum cycle size in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163906 Permutation A163356 applied twice.

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A163916 Permutation A163356 applied thrice.

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A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

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PARI

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A163913 Number of integers i in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356 with A163915(i)=i, but not A163355(i)=i.

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A163355 Permutation of integers for constructing Hilbert curve in N x N grid.

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Maple

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A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

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C

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