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 1-3 of 3 results.

A357000 Number of non-isomorphic cyclic Haar graphs on 2*n nodes.

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 9, 22, 21, 44, 29, 157, 73, 244, 367, 649, 521, 2624, 1609, 7385, 8867, 19400, 16769, 92529, 67553, 216274, 277191, 815557, 662369, 4500266, 2311469
Offset: 1

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Author

Pontus von Brömssen, Sep 08 2022

Keywords

Comments

The first value of n for which a(n) < A002729(n) - 1 is n = 8. This is because the first counterexample to the bicirculant analog to Ádám's conjecture occurs for n = 8. In the terminology of Hladnik, Marušič, and Pisanski, the smallest integer pair (i,j) such that i and j are Haar equivalent (i.e., the cyclic Haar graphs with indices i and j are isomorphic) but not cyclically equivalent (see A357005) is (141,147). See also A357001 and A357002.
Terms a(1)-a(29) were found by generating the cyclic Haar graphs with indices in A333764, and filtering out isomorphic graphs using Brendan McKay's software nauty.

Links

  • Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.
  • Eric Weisstein's World of Mathematics, Haar Graph

Crossrefs

Cf. A002729, A008965, A049287, A091696, A137706, A291166, A333764, A339502, A357001, A357002, A357003, A357004, A357005, A357006.

Formula

a(n) is the number of terms k of A137706 in the interval 2^(n-1) <= k < 2^n.
a(n) is the number of fixed points k of A357004 in the interval 2^(n-1) <= k < 2^n.
a(n) <= A002729(n)-1 <= A091696(n) <= A008965(n).

Extensions

a(30) from Eric W. Weisstein, Jun 27 2023
a(31) from Eric W. Weisstein, Jun 28 2023

A357004 Smallest k for which the cyclic Haar graphs with indices k and n are isomorphic.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 11, 9, 11, 11, 15, 16, 17, 17, 19, 17, 19, 19, 23, 17, 19, 19, 23, 19, 23, 23, 31, 32, 33, 34, 35, 36, 37, 37, 39, 34, 37, 42, 43, 37, 45, 43, 47, 33, 35, 37, 39, 37, 43, 45, 47, 35, 39, 43, 47, 39, 47, 47, 63, 64, 65, 65, 67, 65
Offset: 1

Views

Author

Pontus von Brömssen, Sep 08 2022

Keywords

Comments

The fixed points are the terms of A137706.
The number of fixed points n in the interval 2^(m-1) <= n < 2^m equals A357000(m).

Links

Crossrefs

Cf. A137706 (fixed points; or ordered list of distinct terms), A163382, A272919 (terms that occur only once), A357000, A357005.

Formula

a(a(n)) = a(n).
a(n) = A357005(n) for n <= 146, but a(147) = 141 < 147 = A357005(147).
a(n) <= A357005(n) <= A163382(n).

A357006 Numbers k that are the smallest of all numbers that are cyclically equivalent to k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 19, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 47, 63, 64, 65, 67, 69, 71, 75, 79, 95, 127, 128, 129, 130, 131, 133, 135, 136, 137, 138, 139, 141, 143, 147, 151, 153, 155, 159, 170, 171, 175, 187, 191, 255, 256
Offset: 1

Views

Author

Pontus von Brömssen, Sep 08 2022

Keywords

Comments

For the definition of cyclic equivalence, see A357005, or Hladnik, Marušič, and Pisanski (2002).
The sequence consists of the fixed points of A357005.
The number of terms k in the interval 2^(m-1) <= k < 2^m equals A002729(m)-1.

Links

Crossrefs

Cf. A002729, A137706 (subsequence), subsequence of A333764, A357005.
Showing 1-3 of 3 results.

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