cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 23 results. Next

A163356 Inverse permutation to A163355, related to Hilbert's curve in N x N grid.

Original entry on oeis.org

0, 1, 3, 2, 8, 10, 11, 9, 12, 14, 15, 13, 7, 6, 4, 5, 16, 18, 19, 17, 20, 21, 23, 22, 28, 29, 31, 30, 27, 25, 24, 26, 48, 50, 51, 49, 52, 53, 55, 54, 60, 61, 63, 62, 59, 57, 56, 58, 47, 46, 44, 45, 39, 37, 36, 38, 35, 33, 32, 34, 40, 41, 43, 42, 128, 130, 131, 129, 132, 133
Offset: 0

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Author

Antti Karttunen, Jul 29 2009

Keywords

Links

Crossrefs

Inverse: A163355.
Second and third "powers": A163906, A163916. See also A059252-A059253.
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.
Cf. A059252, A059253, A059905, A059906.
Cf. also A302844, A302846, A302781.

Programs

  • PARI
    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

Formula

a(0) = 0, and provided that d=1, 2 or 3, then a((d*(4^i))+r) = (((2+(i mod 2))^d mod 5)-1) * [either A024036(i) - a(r), if d is 3, and A057300(a(r)) in other cases].
From Antti Karttunen, Apr 14 2018: (Start)
A059905(a(n)) = A059253(n).
A059906(a(n)) = A059252(n).
a(n) = A000695(A059253(n)) + 2*A000695(A059252(n)).
(End)

Extensions

Links to further derived sequences and a nicer Scheme function & formula added by Antti Karttunen, Sep 21 2009

A163890 Orbit size of n in permutation A163355.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 6, 3, 1, 3, 3, 3, 1, 1, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 8, 8, 4, 4, 2, 2, 4, 2, 2, 2, 2, 8, 8, 4, 4, 4, 4, 8, 8, 4, 2, 4, 2, 4, 8, 4, 8, 2, 2, 2, 2, 6, 6, 6, 6, 12, 6, 6, 12, 12, 12, 6, 6, 18, 18, 18, 6, 3, 3, 6, 6, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 6, 3, 9, 18
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Comments

Will all terms of A003586 eventually appear?

Links

Crossrefs

See A163891, A163892, A163894.

Programs

  • Scheme
    (define (A163890 n) (let loop ((i 1) (nn (A163355 n))) (cond ((= nn n) i) (else (loop (1+ i) (A163355 nn))))))

A163904 Array A(i,j): Cycle size of each A054238(i,j) in permutation A163355, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

1, 1, 2, 6, 2, 6, 3, 3, 6, 3, 1, 6, 6, 3, 4, 3, 3, 3, 6, 8, 8, 1, 3, 4, 3, 4, 4, 2, 1, 2, 2, 2, 2, 2, 2, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 6, 6, 2, 4, 4, 8, 8, 4, 6, 6, 12, 6, 12, 2, 4, 8, 4, 4, 12, 6, 18, 6, 6, 12, 6, 2, 4, 8, 4, 6, 12, 18, 6, 3, 12, 18, 6, 9, 2, 2, 8, 6, 6, 12, 18, 3, 3, 6, 18, 18
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Examples

			The top left 8x8 corner of this array:
1 1 6 3 1 3 1 1
2 2 3 6 3 3 2 2
6 6 6 3 4 2 4 2
3 3 6 3 2 4 4 2
4 8 4 2 4 4 4 2
8 4 2 4 8 8 4 2
2 2 8 8 4 8 2 2
2 2 4 4 4 8 2 2

Links

Crossrefs

a(n) = A163890(A054238(n)) = A163890(A163357(n)). Positions of 1's: A165403. See also A163898, A163899.

A163901 The positions i where A163355(i) = i, that is, the fixed points of permutation A163355.

Original entry on oeis.org

0, 1, 16, 20, 21, 256, 260, 261, 320, 321, 336, 340, 341, 4096, 4100, 4101, 4160, 4161, 4176, 4180, 4181, 5120, 5121, 5136, 5140, 5141, 5376, 5380, 5381, 5440, 5441, 5456, 5460, 5461, 65536, 65540, 65541, 65600, 65601, 65616, 65620, 65621
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Comments

Alternatively, the positions of zeros in A163900.

Crossrefs

a(n) = A000695(A165404(n)). Same sequence in base-4: A165406. See also A163902, A163903, A163910.

A163905 Permutation A163355 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 9, 10, 11, 8, 14, 12, 15, 13, 4, 6, 7, 5, 16, 18, 19, 17, 20, 21, 22, 23, 27, 25, 26, 24, 30, 29, 28, 31, 47, 45, 44, 46, 39, 37, 38, 36, 40, 41, 42, 43, 33, 34, 35, 32, 58, 56, 57, 59, 54, 53, 52, 55, 49, 51, 48, 50, 60, 61, 62, 63, 149, 150, 151, 148, 152, 154
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Links

Crossrefs

Inverse: A163906. a(n) = A163355(A163355(n)). Array A163907 shows this in N x N grid. See also A163915.

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

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Crossrefs

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

A163915 Permutation A163355 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, 28, 29, 31, 30, 27, 25, 24, 26, 48, 51, 50, 49, 52, 53, 55, 54, 60, 61, 63, 62, 57, 59, 56, 58, 47, 46, 45, 44, 39, 37, 36, 38, 35, 34, 32, 33, 40, 41, 43, 42, 127, 125, 126, 124, 118, 117
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Links

Crossrefs

Inverse: A163916. Array A163917 shows this in N X N grid. See also A163905.
Cf. A163355

Formula

a(n) = A163355(A163355(A163355(n))) = A163355(A163905(n)).

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

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Crossrefs

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

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Crossrefs

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

A163894 The least i for which A163355^n(i) is not equal to i, 0 if no such i exists, i.e., when A163355^n = A001477.

Original entry on oeis.org

0, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 33, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 76, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 33, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 76, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 33, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 390, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4, 2, 33, 2, 4, 2, 4, 2, 24, 2, 4, 2, 4
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Comments

A163355^n means n-fold application of A163355, i.e., A163355^2 = A163905, A163355^3 = A163915. By convention A163355^0 = A001477.

Links

Crossrefs

See A163890, A163895, A163896.

Programs

  • Maple
    A163894 := proc(n)
    local i,a355,a,itr ;
    if n = 0 then
        return 0 ;
    end if;
    a := 0 ;
    for i from 0 do
        a355 := A163355(i) ;
        for itr from 2 to n do
            a355 := A163355(a355) ;
        end do:
        if a355 <> i then
            return i ;
        end if;
    end do:
end proc:
seq(A163894(n),n=0..100) ; # R. J. Mathar, Nov 22 2023
Showing 1-10 of 23 results. Next

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