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 1-4 of 4 results.

A213338 Costas arrays such that the corresponding permutation is cyclic.

Original entry on oeis.org

1, 1, 2, 2, 10, 26, 32, 74, 54, 198, 486, 726, 1112, 1438, 1570, 1576, 1220, 954, 888, 464, 194, 116, 48, 8, 0, 0, 0, 36, 0
Offset: 1

Views

Author

Joerg Arndt, Jun 09 2012

Keywords

Links

Crossrefs

Cf. A008404 (Costas arrays), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213339 (Costas arrays that are connected).

A213270 Costas arrays such that the corresponding permutation is an involution.

Original entry on oeis.org

1, 2, 2, 2, 4, 10, 20, 18, 20, 28, 36, 34, 50, 46, 62, 40, 38, 20, 12, 8, 16, 10, 20, 0, 4, 4, 14, 0, 10
Offset: 1

Views

Author

Joerg Arndt, Jun 08 2012

Keywords

Comments

Self-inverse permutations such that each row in the difference table consists of pairwise distinct elements (see example).

Examples

			The permutation (4, 7, 9, 1, 6, 5, 2, 8, 3) is an involution and corresponds to a Costas array:
   4  7  9  1  6  5  2  8  3  (Permutation: p(1), p(2), p(3), ..., p(n) )
   3  2 -8  5 -1 -3  6 -5     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
   5 -6 -3  4 -4  3  1        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -3 -1 -4  1  2 -2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
   2 -2 -7  7 -3              ( etc. )
   1 -5 -1  2
  -2  1 -6
   4 -4
  -1

Links

Crossrefs

Cf. A008404 (Costas arrays), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).

A213339 Costas arrays such that the corresponding permutation is connected.

Original entry on oeis.org

1, 1, 2, 6, 26, 80, 152, 348, 628, 1868, 3870, 7014, 11788, 15746, 18388, 19820, 17218, 14344, 9844, 6238, 3430, 1968, 814, 184, 84, 52, 190, 656, 132
Offset: 1

Views

Author

Joerg Arndt, Jun 09 2012

Keywords

Links

Crossrefs

Cf. A008404 (Costas arrays), A003319 (connected permutations), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic).

A213272 Costas arrays such that the terms in each row of the difference table are unique modulo n.

Original entry on oeis.org

1, 2, 0, 8, 0, 12, 0, 0, 0, 40, 0, 48, 0, 0, 0, 128, 0, 108, 0, 0, 0, 220, 0, 0, 0, 0, 0, 336, 0
Offset: 1

Views

Author

Joerg Arndt, Jun 08 2012

Keywords

Comments

Permutations of n elements such that each row in the difference table consists of pairwise distinct elements, even when taken modulo n (see example).
For n<=29 the nonzero terms a(n) appear for n in A006093 (primes minus 1) and a(n)=A002618(n) (n*phi(n)); omitting the zeros we obtain A104039 (number of primitive roots modulo (p(n))^2, where p(n) is n-th prime).
A002618(n) divides a(n) for all n, since (treating elements as integers modulo n) adding or subtracting a constant from each element or multiplying each element by an integer coprime to n preserves distinctness of all values modulo n. - Charlie Neder, May 26 2019

Examples

			The permutation (10, 9, 2, 8, 6, 1, 3, 7, 4, 5) corresponds to a Costas array:
  10  9  2  8  6  1  3  7  4  5  (Permutation: p(1), p(2), p(3), ..., p(n) )
  -1 -7  6 -2 -5  2  4 -3  1     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
  -8 -1  4 -7 -3  6  1 -2        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -2 -3 -1 -5  1  3  2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
  -4 -8  1 -1 -2  4              ( etc. )
  -9 -6  5 -4 -1
  -7 -2  2 -3
  -3 -5  3
  -6 -4
  -5
The values in each row are unique also modulo n=10:
  10 9 2 8 6 1 3 7 4 5  (Permutation: p(1), p(2), p(3), ..., p(n) )
   9 3 6 8 5 2 4 7 1    (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
   2 9 4 3 7 6 1 8      (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
   8 7 9 5 1 3 2        (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
   6 2 1 9 8 4          ( etc. )
   1 4 5 6 9
   3 8 2 7
   7 5 3
   4 6
   5

Links

Crossrefs

Cf. A008404 (Costas arrays), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).
Showing 1-4 of 4 results.

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