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 1-9 of 9 results.

A163891 Positions where A163890 obtains distinct new values.

Original entry on oeis.org

0, 2, 4, 5, 24, 33, 68, 76, 96, 390, 536, 561, 1092, 1093, 1156, 1220, 1554, 6242, 8465, 8584, 8977, 17953, 18500, 19564, 99396, 99873, 101444, 137286, 143633, 279620, 279622, 287108, 287248, 397585, 401796, 1597720, 1597969, 1598532
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Links

Crossrefs

See A163892, A163893, A163896, A163911, A163912.

A163892 Distinct values in A163890 in the order of appearance.

Original entry on oeis.org

1, 2, 6, 3, 4, 8, 12, 18, 9, 16, 32, 24, 108, 36, 48, 54, 72, 64, 96, 192, 216, 324, 144, 162, 288, 576, 648, 384, 1944, 432, 864, 486, 972, 1296, 27, 768, 1152, 128, 5832, 1728, 1536, 3456, 2304
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Comments

Probably a permutation of A003586 (A036561).

Crossrefs

a(n) = A163890(A163891(n)). See also A163891, A163893, A163894, A163912.

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

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Links

Crossrefs

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.

Programs

  • Maple
    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
  • 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); };
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

Formula

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)

Extensions

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

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.

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.

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

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

Views

Author

Antti Karttunen, Sep 19 2009

Keywords

Crossrefs

a(n) = A163913(n)/3. Bisections: A163909, A163919. See also A163903, A163911, A163912, A163904, A163890.
Showing 1-9 of 9 results.

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