A227005 -id:A227005 - cp's OEIS frontend

A063524 Characteristic function of 1.

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Labos Elemer, Jul 30 2001

The identity function for Dirichlet multiplication (see Apostol).
Sum of the Moebius function mu(d) of the divisors d of n. - Robert G. Wilson v, Sep 30 2006
-a(n) is the Hankel transform of A000045(n), n >= 0 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
a(A000012(n)) = 1; a(A087156(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
a(n) for n >= 1 is the Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A008683(n) * A000012(n), a(n) = A007427(n) * A000005(n), a(n) = A007428(n) * A007425(n). - Jaroslav Krizek, Mar 03 2009
From Christopher Hunt Gribble, Jul 11 2013: (Start)
a(n) for 1 <= n <= 4 and conjectured for n > 4 is the number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element: When n=1, there is only 1 Hamiltonian circuit in a 2 X 2 square lattice, as illustrated below.  The circuit is the same when rotated and/or reflected and so has only 1 orbital element under the symmetry group of the square.
     o--o
     |  |
     o--o  (End)
Convolution property: For any sequence b(n), the sequence c(n)=b(n)*a(n) has the following values: c(1)=0, c(n+1)=b(n) for all n > 1. In other words, the sequence b(n) is shifted 1 step to the right. - David Neil McGrath, Nov 10 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

Antti Karttunen, Table of n, a(n) for n = 0..100000
G. P. Michon, Multiplicative Functions
Wikipedia, Dirichlet convolution
Index entries for sequences computed from exponents in factorization of n
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for characteristic functions

Cf. A000007 (same sequence shifted once left).

Cf. A000005, A000010, A000012, A000027, A003763, A008683, A007427, A007428, A007425, A023900, A055615, A101296, A227005, A227257, A227301, A209077.

a063524 = fromEnum . (== 1)  -- Reinhard Zumkeller, Apr 01 2012

A063524 := proc(n) if n = 1 then 1 else 0; fi; end;

Table[If[n == 1, 1, 0], {n, 0, 104}] (* Robert G. Wilson v, Sep 30 2006 *)
LinearRecurrence[{1},{0,1,0},106] (* Ray Chandler, Jul 15 2015 *)

a(n)=n==1; \\ Charles R Greathouse IV, Apr 01 2012

def A063524(n): return int(n==1) # Chai Wah Wu, Feb 04 2022

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: x.
E.g.f.: x. (End)
a(n) = mu(n^2). - Enrique Pérez Herrero, Sep 04 2009
a(n) = floor(n/A000203(n)) for n > 0. - Enrique Pérez Herrero, Nov 11 2009
a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - Luce ETIENNE, Jun 05 2015
a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - Chayim Lowen, Aug 01 2015
a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - Chayim Lowen, Aug 05 2015
From Antti Karttunen, Jun 04 2022: (Start)
For n >= 1:
a(n) = Sum_{d|n} A000010(n/d) * A023900(d), and similarly for any pair of sequences that are Dirichlet inverses of each other, like for example A000027 & A055615 and those mentioned in Krizek's Mar 03 2009 comment above.
a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.
Fully multiplicative with a(p^e) = 0. (End)

A227257 Number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

Original entry on oeis.org

0, 1, 24, 1760, 411861, 551247139, 2883245852086, 85948329517780776, 11001968794030973784902, 7462399462450938863305238264

Offset: 1

Christopher Hunt Gribble, Jul 05 2013

			When n = 2, there is only 1 Hamiltonian circuit in a 4 X 4 square lattice, where the orbits under the symmetry group of the square have 4 elements.  The 4 elements are:
    o__o__o__o    o__o__o__o    o__o__o__o    o__o  o__o
    |        |    |        |    |        |    |  |  |  |
    o  o__o__o    o  o__o  o    o__o__o  o    o  o  o  o
    |  |          |  |  |  |          |  |    |  |  |  |
    o  o__o__o    o  o  o  o    o__o__o  o    o  o__o  o
    |        |    |  |  |  |    |        |    |        |
    o__o__o__o    o__o  o__o    o__o__o__o    o__o__o__o

Giovanni Resta, Simple C program for computing a(1)-a(4)
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A003763, A209077, A063524, A227005, A227301.

A063524 + A227005 + A227257 + A227301 = A209077.
1*A063524 + 2*A227005 + 4*A227257 + 8*A227301 = A003763.
a(n) = A237429(n) + A237430(n). - Ed Wynn, Feb 07 2014

a(4) from Giovanni Resta, Jul 11 2013

a(5)-a(10) from Ed Wynn, Feb 05 2014

A227301 Number of Hamiltonian circuits in a 2n node X 2n node square lattice, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 8 elements.

Original entry on oeis.org

0, 0, 121, 578937, 58407351059, 134528360800075421, 7015812452559988037073365, 8235314565328229497795808499821534, 216740797236120772990968348272561831275923059, 127557553423846099192878370706037904215158660401579043097

Offset: 1

Christopher Hunt Gribble, Jul 05 2013

			When n = 3 there are 121 Hamiltonian circuits in a 6 X 6  square lattice where the orbits under the symmetry group of the square have 8 elements.  One of these circuits is shown below with its 8 distinct transformations under rotation and reflection:
o__o__o__o__o__o    o__o  o__o  o__o    o__o__o__o__o__o
|              |    |  |  |  |  |  |    |              |
o__o__o__o  o__o    o  o  o  o  o  o    o__o__o  o__o__o
         |  |       |  |  |  |  |  |          |  |
o__o__o__o  o__o    o  o__o  o  o  o    o__o__o  o__o__o
|              |    |        |  |  |    |              |
o__o__o  o__o__o    o  o__o  o__o  o    o__o  o__o__o__o
      |  |          |  |  |        |       |  |
o__o__o  o__o__o    o  o  o  o__o  o    o__o  o__o__o__o
|              |    |  |  |  |  |  |    |              |
o__o__o__o__o__o    o__o  o__o  o__o    o__o__o__o__o__o
.
o__o  o__o  o__o    o__o__o__o__o__o    o__o  o__o  o__o
|  |  |  |  |  |    |              |    |  |  |  |  |  |
o  o__o  o  o  o    o__o  o__o__o__o    o  o  o  o__o  o
|        |  |  |       |  |             |  |  |        |
o  o__o  o__o  o    o__o  o__o__o__o    o  o__o  o__o  o
|  |  |        |    |              |    |        |  |  |
o  o  o  o__o  o    o__o__o  o__o__o    o  o__o  o  o  o
|  |  |  |  |  |          |  |          |  |  |  |  |  |
o  o  o  o  o  o    o__o__o  o__o__o    o  o  o  o  o  o
|  |  |  |  |  |    |              |    |  |  |  |  |  |
o__o  o__o  o__o    o__o__o__o__o__o    o__o  o__o  o__o
.
o__o__o__o__o__o    o__o  o__o  o__o
|              |    |  |  |  |  |  |
o__o__o  o__o__o    o  o  o  o  o  o
      |  |          |  |  |  |  |  |
o__o__o  o__o__o    o  o  o  o__o  o
|              |    |  |  |        |
o__o__o__o  o__o    o  o__o  o__o  o
         |  |       |        |  |  |
o__o__o__o  o__o    o  o__o  o  o  o
|              |    |  |  |  |  |  |
o__o__o__o__o__o    o__o  o__o  o__o

Giovanni Resta, Simple C program for computing a(1)-a(4)
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A003763, A209077, A063524, A227005, A227257.

A063524 + A227005 + A227257 + A227301 = A209077.

1*A063524 + 2*A227005 + 4*A227257 + 8*A227301 = A003763.

a(4) from Giovanni Resta, Jul 11 2013

a(5)-a(10) from Ed Wynn, Feb 05 2014

A237431 Number of nonisomorphic Hamiltonian cycles on 2n X 2n square grid of points with exactly two axes of reflective symmetry.

Original entry on oeis.org

0, 1, 3, 20, 244, 6891, 378813, 47917598, 12118420172, 6998287399637

Offset: 1

Ed Wynn, Feb 07 2014

			Examples of 2 of the 3 classes for n=3. Note that all examples also have two-fold (but not four-fold) rotational symmetry.
  o-o-o-o-o-o   o-o-o-o-o-o
  |         |   |         |
  o-o-o o-o-o   o o-o o-o o
      | |       | | | | | |
  o-o-o o-o-o   o-o o o o-o
  |         |       | |
  o-o-o o-o-o   o-o o o o-o
      | |       | | | | | |
  o-o-o o-o-o   o o-o o-o o
  |         |   |         |
  o-o-o-o-o-o   o-o-o-o-o-o

Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A209077, A227005, A237432.

A237432 Number of nonisomorphic Hamiltonian cycles on (4n-2) X (4n-2) square grid of points with four-fold rotational symmetry (and no other symmetry).

Original entry on oeis.org

0, 1, 102, 255359, 15504309761, 21955745395591600, 712319733455900182066337, 524246290066954425217045809870657

Offset: 1

Ed Wynn, Feb 07 2014

For square grids of m X m points, there are solutions only for m = (4n-2).

			The two cycles counted as a single class for n=2. These are isomorphic (here meaning isomorphic under the full symmetry group of the square), since each is a reflection of the other.
  o-o o-o-o-o  o-o-o-o o-o
  | | |     |  |     | | |
  o o o o-o-o  o-o-o o o o
  | | | |          | | | |
  o o-o o-o-o  o-o-o o-o o
  |         |  |         |
  o-o-o o-o o  o o-o o-o-o
      | | | |  | | | |
  o-o-o o o o  o o o o-o-o
  |     | | |  | | |     |
  o-o-o-o o-o  o-o o-o-o-o

Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A209077, A227005, A237431.

a(n) = A238819(n-1) / 2 for n > 1. - Andrew Howroyd, Apr 06 2016

a(6)-a(8) from Andrew Howroyd, Apr 06 2016

cp's OEIS Frontend

A063524 Characteristic function of 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A227257 Number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A227301 Number of Hamiltonian circuits in a 2n node X 2n node square lattice, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 8 elements.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A237431 Number of nonisomorphic Hamiltonian cycles on 2n X 2n square grid of points with exactly two axes of reflective symmetry.

Views

Author

Keywords

Examples

Links

Crossrefs

A237432 Number of nonisomorphic Hamiltonian cycles on (4n-2) X (4n-2) square grid of points with four-fold rotational symmetry (and no other symmetry).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions