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 14 results. Next

A163333 Self-inverse permutation of integers: A163327-conjugate of A163332.

Original entry on oeis.org

0, 7, 2, 3, 4, 5, 6, 1, 8, 69, 64, 71, 66, 67, 68, 63, 70, 65, 18, 25, 20, 21, 22, 23, 24, 19, 26, 29, 34, 27, 32, 31, 30, 35, 28, 33, 44, 37, 42, 41, 40, 39, 38, 43, 36, 47, 52, 45, 50, 49, 48, 53, 46, 51, 54, 61, 56, 57, 58, 59, 60, 55, 62, 15, 10, 17, 12, 13, 14, 9, 16, 11
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Comments

The integers [0,(9^k)-1] are confined to range [0,(9^k)-1].

Links

Crossrefs

a(n) = A163327(A163332(A163327(n))). A163334 & A163336 give two variants of the Peano curve in an N x N grid. Cf. also A163355.

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, 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

0, 1, 5, 2, 4, 6, 15, 3, 7, 47, 16, 14, 8, 46, 48, 17, 13, 9, 45, 49, 53, 18, 12, 10, 44, 50, 52, 54, 19, 23, 11, 43, 39, 51, 55, 59, 20, 22, 24, 42, 40, 38, 56, 58, 60, 141, 21, 25, 29, 41, 37, 69, 57, 61, 425, 142, 140, 26, 28, 30, 36, 70, 68, 62, 424, 426, 143, 139
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Examples

			The top left 9 X 9 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  1  2 15 16 17 18 19 20
   5  4  3 14 13 12 23 22 21
   6  7  8  9 10 11 24 25 26
  47 46 45 44 43 42 29 28 27
  48 49 50 39 40 41 30 31 32
  53 52 51 38 37 36 35 34 33
  54 55 56 69 70 71 72 73 74
  59 58 57 68 67 66 77 76 75
  60 61 62 63 64 65 78 79 80

Links

Crossrefs

Transpose: A163336. Inverse: A163335. One-based version: A163338. Row sums: A163342. Row 0: A163480. Column 0: A163481. Central diagonal: A163343.
See also: A163528-A163529, A163531, A163534, A163536, A163897.
See A163357 and A163359 for the Hilbert curve.

Programs

  • Mathematica
    b[{n_, k_}, {m_}] := (A[k, n] = m - 1);
MapIndexed[b, List @@ PeanoCurve[4][[1]]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

Formula

a(n) = A163332(A163328(n)).

Extensions

Links to further derived sequences added by Antti Karttunen, Sep 21 2009
Name corrected by Kevin Ryde, Aug 22 2020

A163336 Peano curve in an n X n grid, starting downwards from the top left corner, 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

0, 5, 1, 6, 4, 2, 47, 7, 3, 15, 48, 46, 8, 14, 16, 53, 49, 45, 9, 13, 17, 54, 52, 50, 44, 10, 12, 18, 59, 55, 51, 39, 43, 11, 23, 19, 60, 58, 56, 38, 40, 42, 24, 22, 20, 425, 61, 57, 69, 37, 41, 29, 25, 21, 141, 426, 424, 62, 68, 70, 36, 30, 28, 26, 140, 142, 431, 427
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Examples

			The top left 9 X 9 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  5  6 47 48 53 54 59 60
   1  4  7 46 49 52 55 58 61
   2  3  8 45 50 51 56 57 62
  15 14  9 44 39 38 69 68 63
  16 13 10 43 40 37 70 67 64
  17 12 11 42 41 36 71 66 65
  18 23 24 29 30 35 72 77 78
  19 22 25 28 31 34 73 76 79
  20 21 26 27 32 33 74 75 80

Links

Crossrefs

Transpose: A163334. Inverse: A163337. a(n) = A163332(A163330(n)) = A163327(A163333(A163328(n))) = A163334(A061579(n)). One-based version: A163340. Row sums: A163342. Row 0: A163481. Column 0: A163480. Central diagonal: A163343.
See A163357 and A163359 for the Hilbert curve.

Programs

  • Mathematica
    b[{n_, k_}, {m_}] := (A[n, k] = m - 1);
MapIndexed[b, List @@ PeanoCurve[4][[1]]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

Extensions

Name corrected by Kevin Ryde, Aug 28 2020

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

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 01 2009

Keywords

Comments

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Links

Crossrefs

Cf. A025581, A163335, A002262, A163337, A163325, A163332.
See also: A059252, A059253, A163529, A163530, A163531, A163532.

Formula

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Extensions

Name corrected by Kevin Ryde, Aug 28 2020

A163529 The Y-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8
Offset: 0

Views

Author

Antti Karttunen, Aug 01 2009

Keywords

Comments

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163529.pdf. - Jeffrey Shallit, Aug 10 2023

Links

Crossrefs

Cf. A002262, A163335, A025581, A163337, A163326, A163332.
Cf. A059252, A059253, A163528, A163530, A163531, A163533.

Formula

a(n) = A002262(A163335(n)) = A025581(A163337(n)) = A163326(A163332(n)).

Extensions

Name corrected by Kevin Ryde, Aug 28 2020

A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 9, 5, 7, 27, 10, 12, 8, 28, 30, 11, 13, 15, 29, 31, 33, 18, 14, 16, 36, 32, 34, 54, 19, 21, 17, 37, 39, 35, 55, 57, 20, 22, 24, 38, 40, 42, 56, 58, 60, 81, 23, 25, 45, 41, 43, 63, 59, 61, 243, 82, 84, 26, 46, 48, 44, 64, 66, 62, 244, 246, 83, 85, 87, 47, 49
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Examples

			From _Kevin Ryde_, Oct 06 2020: (Start)
Array A(y,x) read by downwards antidiagonals, so 0, 1,3, 2,4,6, etc.
        x=0   1   2   3   4   5   6   7   8
      +--------------------------------------
  y=0 |   0,  1,  2,  9, 10, 11, 18, 19, 20,
    1 |   3,  4,  5, 12, 13, 14, 21, 22,
    2 |   6,  7,  8, 15, 16, 17, 24,
    3 |  27, 28, 29, 36, 37, 38,
    4 |  30, 31, 32, 39, 40,
    5 |  33, 34, 35, 42,
    6 |  54, 55, 56,
    7 |  57, 58,
    8 |  60,
(End)

Links

Crossrefs

Inverse: A163329. Transpose: A163330. Cf. A037314 (row y=0), A208665 (column x=0)
Cf. A054238 is an analogous sequence for binary. Cf. A007089, A163327, A163332, A163334.

Programs

Formula

a(n) = A037314(A025581(n)) + 3*A037314(A002262(n))
a(n) = A163327(A163330(n)).

Extensions

Edited by Charles R Greathouse IV, Nov 01 2009

A163335 Inverse permutation to A163334.

Original entry on oeis.org

0, 1, 3, 7, 4, 2, 5, 8, 12, 17, 23, 30, 22, 16, 11, 6, 10, 15, 21, 28, 36, 46, 37, 29, 38, 47, 57, 69, 58, 48, 59, 70, 82, 96, 83, 71, 60, 50, 41, 32, 40, 49, 39, 31, 24, 18, 13, 9, 14, 19, 25, 33, 26, 20, 27, 34, 42, 52, 43, 35, 44, 53, 63, 74, 86, 99, 85, 73, 62, 51, 61, 72
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Comments

abs(A025581(a(n+1)) - A025581(a(n))) + abs(A002262(a(n+1)) - A002262(a(n))) = 1 for all n.

Links

Crossrefs

Inverse: A163334. a(n) = A163329(A163332(n)). One-based version: A163339. See also A163337, A163358.

A163337 Inverse permutation to A163336.

Original entry on oeis.org

0, 2, 5, 8, 4, 1, 3, 7, 12, 18, 25, 33, 26, 19, 13, 9, 14, 20, 27, 35, 44, 53, 43, 34, 42, 52, 63, 74, 62, 51, 61, 73, 86, 99, 85, 72, 60, 49, 39, 31, 40, 50, 41, 32, 24, 17, 11, 6, 10, 16, 23, 30, 22, 15, 21, 29, 38, 47, 37, 28, 36, 46, 57, 69, 82, 96, 83, 70, 58, 48, 59, 71
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Comments

abs(A025581(a(n+1))-A025581(a(n))) + abs(A002262(a(n+1))-A002262(a(n))) = 1 for all n.

Links

Crossrefs

Inverse: A163336. a(n) = A163331(A163332(n)) = A061579(A163335(n)). One-based version: A163341. See also A163335, A163358.

A163330 Square array A, where entry A(y,x) has the ternary digits of y interleaved with the ternary digits of x, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 3, 1, 6, 4, 2, 27, 7, 5, 9, 30, 28, 8, 12, 10, 33, 31, 29, 15, 13, 11, 54, 34, 32, 36, 16, 14, 18, 57, 55, 35, 39, 37, 17, 21, 19, 60, 58, 56, 42, 40, 38, 24, 22, 20, 243, 61, 59, 63, 43, 41, 45, 25, 23, 81, 246, 244, 62, 66, 64, 44, 48, 46, 26, 84, 82, 249, 247, 245
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Links

Crossrefs

Inverse: A163331. a(n) = A163327(A163328(n)). Transpose: A163328. Cf. A007089, A163327, A163332, A163334.

Programs

Formula

a(n) = 3*A037314(A025581(n)) + A037314(A002262(n))
Showing 1-10 of 14 results. Next

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