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 1-7 of 7 results.

A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 1, 6, 6, 5, 6, 2, 8, 8, 6, 3, 8, 4, 10, 10, 7, 8, 1, 10, 6, 12, 12, 8, 5, 10, 2, 12, 8, 14, 14, 9, 10, 3, 12, 4, 14, 10, 16, 16, 10, 7, 12, 1, 14, 6, 16, 12, 18, 18, 11, 12, 5, 14, 2, 16, 8, 18, 14, 20, 20, 12, 9, 14, 3, 16, 4, 18, 10, 20, 16, 22, 22, 13, 14, 7, 16, 1
Offset: 0

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Comments

Simple periodic site swap permutations of natural numbers.
Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...

Examples

			Table begins:
1 2 3 4 5 6 7 ...
2 4 1 6 3 8 5 ...
4 6 2 8 1 10 3 ...
6 8 4 10 2 12 1 ...

Links

Crossrefs

Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.

Programs

  • Maple
    PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
[seq(PerSS_table(j),j=0..119)];

Formula

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).

A084455 Permutation of Z, obtained by reflecting the juggling sequence A084452 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 10, 2, 12, 4, 14, 6, 8, 9, 20, 3, 22, 5, 24, 7, 16, 17, 18, 19, 32, 11, 28, 13, 34, 15, 26, 27, 38, 23, 30, 31, 44, 21, 46, 25, 36, 37, 50, 29, 40, 41, 42, 43, 58, 33, 52, 35, 48, 49, 62, 39, 64, 47, 54, 55, 56, 57, 72, 45, 60, 61, 74, 51, 76, 53, 66, 67, 68, 69, 70, 71, 86
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

This permutation consists of three infinite cycles + infinite number of fixed points.

Links

Crossrefs

Inverse: A084466. Cf. also A084454, A084461, A084491, A084495, A084499, A065166.

Programs

  • Maple
    [seq(Z2N(A084455_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084455_Z2Z := z -> z+`if`((z > 0), A084452(z),`if`((z >= -3),2*(-z), A084452(A084454((-z)-3))));

A084461 Permutation of Z, obtained by reflecting the juggling sequence A084458 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 10, 2, 12, 4, 18, 6, 8, 9, 20, 3, 28, 5, 14, 15, 16, 17, 24, 7, 34, 11, 22, 23, 38, 19, 26, 27, 46, 13, 30, 31, 32, 33, 48, 21, 36, 37, 58, 25, 40, 41, 42, 43, 44, 45, 60, 29, 72, 35, 50, 51, 52, 53, 54, 55, 56, 57, 66, 39, 80, 47, 62, 63, 64, 65, 84, 59, 68, 69, 70, 71, 94
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

This permutation consists of three infinite cycles + infinite number of fixed points.

Links

Crossrefs

Inverse: A084462. Cf. also A084460, A084455, A084491, A084495, A084499, A065166.

Programs

  • Maple
    [seq(Z2N(A084461_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084461_Z2Z := z -> z+`if`((z > 0), A084458(z),`if`((z >= -3),2*(-z), A084458(A084460((-z)-3))));

A084491 Permutation of Z, obtained by reflecting the juggling sequence A084501 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 8, 2, 10, 4, 12, 6, 16, 3, 14, 5, 18, 7, 20, 11, 22, 9, 24, 13, 28, 15, 26, 17, 32, 19, 30, 23, 34, 21, 38, 27, 40, 25, 36, 29, 46, 35, 42, 31, 44, 33, 52, 39, 50, 41, 48, 37, 54, 47, 56, 45, 58, 43, 60, 49, 62, 51, 64, 53, 68, 55, 66, 57, 70, 59, 74, 63, 72, 61, 76, 65, 78
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

This permutation consists of three infinite cycles + infinite number of fixed points.

Links

Crossrefs

Inverse: A084492. Cf. also A084490, A084455, A084461, A084495, A084499, A065166.

Programs

  • Maple
    [seq(Z2N(A084491_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084491_Z2Z := z -> z+`if`((z > 0), A084501(z),`if`((z >= -3),2*(-z), A084501(A084490((-z)-3))));

A084495 Permutation of Z, obtained by reflecting the juggling sequence A084511 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 8, 2, 12, 4, 10, 6, 16, 3, 18, 7, 14, 5, 24, 13, 20, 9, 22, 11, 30, 17, 28, 19, 26, 15, 34, 25, 36, 23, 38, 21, 32, 33, 42, 27, 46, 29, 40, 31, 44, 39, 50, 35, 54, 41, 52, 37, 48, 49, 60, 43, 56, 47, 62, 45, 58, 53, 68, 57, 66, 51, 70, 55, 64, 65, 76, 61, 78, 59, 72, 63, 74
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

This permutation consists of three infinite cycles + infinite number of fixed points.

Links

Crossrefs

Inverse: A084496. Cf. also A084494, A084455, A084461, A084491, A084499, A065166.

Programs

  • Maple
    [seq(Z2N(A084495_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084495_Z2Z := z -> z+`if`((z > 0), A084511(z),`if`((z >= -3),2*(-z), A084511(A084494((-z)-3))));

A084499 Permutation of Z, obtained by reflecting the juggling sequence A084521 from positive to negative numbers (with zero thrown at beat 0), folded to N with functions N2Z and Z2N.

Original entry on oeis.org

1, 8, 2, 12, 4, 10, 6, 16, 3, 18, 7, 14, 5, 24, 13, 20, 9, 22, 11, 30, 17, 28, 19, 26, 15, 34, 25, 36, 23, 38, 21, 32, 33, 42, 27, 46, 29, 44, 31, 40, 41, 52, 35, 48, 39, 54, 37, 50, 45, 60, 49, 58, 43, 62, 47, 56, 57, 68, 53, 70, 51, 64, 55, 66, 63, 76, 65, 78, 59, 74, 61, 72
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

This permutation consists of three infinite cycles + infinite number of fixed points.

Links

Crossrefs

Inverse: A084530. Cf. also A084498, A084455, A084461, A084491, A084495, A065166.

Programs

  • Maple
    [seq(Z2N(A084499_Z2Z(N2Z(n))),n=1..45)];
N2Z := n -> ((-1)^n)*floor(n/2);
Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
A084499_Z2Z := z -> z+`if`((z > 0), A084511(z),`if`((z >= -3),2*(-z), A084521(A084498((-z)-3))));

A065170 Permutation t->t-3 of Z, folded to N.

Original entry on oeis.org

7, 5, 9, 3, 11, 1, 13, 2, 15, 4, 17, 6, 19, 8, 21, 10, 23, 12, 25, 14, 27, 16, 29, 18, 31, 20, 33, 22, 35, 24, 37, 26, 39, 28, 41, 30, 43, 32, 45, 34, 47, 36, 49, 38, 51, 40, 53, 42, 55, 44, 57, 46, 59, 48, 61, 50, 63, 52, 65, 54, 67, 56, 69, 58, 71, 60, 73, 62, 75, 64, 77, 66
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Comments

This permutation consists of just three cycles, which are infinite.

Links

Crossrefs

Inverse permutation to A065166.

Programs

  • Mathematica
    CoefficientList[Series[(x^8 - x^7 + 4 x^6 - 4 x^5 + 4 x^4 - 4 x^3 - 3 x^2 - 2 x + 7)/((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 08 2014 *)
LinearRecurrence[{1,1,-1},{7,5,9,3,11,1,13,2,15},80] (* Harvey P. Dale, Oct 19 2018 *)
  • PARI
    Vec(x*(x^8-x^7+4*x^6-4*x^5+4*x^4-4*x^3-3*x^2-2*x+7)/((x-1)^2*(x+1))  + O(x^100)) \\ Colin Barker, Mar 07 2014

Formula

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)-3).
G.f.: x*(x^8-x^7+4*x^6-4*x^5+4*x^4-4*x^3-3*x^2-2*x+7) / ((x-1)^2*(x+1)). - Colin Barker, Feb 18 2013
a(n) = -6*(-1)^n+n for n>6. a(n) = a(n-1)+a(n-2)-a(n-3) for n>9. - Colin Barker, Mar 07 2014
Sum_{n>=1} (-1)^n/a(n) = 46/15 - log(2). - Amiram Eldar, Aug 08 2023
Showing 1-7 of 7 results.

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