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A063524 Characteristic function of 1. 63
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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. W. Lang Jan 23 2007.

a(A000012(n)) = 1; a(A087156(n)) = 0. - Reinhard Zumkeller, Oct 11 2008

a(n) for n >= 1 is 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

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  - Christopher Hunt Gribble, Jul 11 2013

REFERENCES

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

LINKS

Table of n, a(n) for n=0..105.

G. P. Michon, Multiplicative Functions.

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for characteristic functions

FORMULA

a(n) = (n!^2 mod (n+1))*((n+1)!^2 mod (n+2)), with n>=0 - Paolo P. Lava, Apr 24 2007

a(n) = C((n+1)^2,n+3) mod 2 = C((n+13)^4,n+15) mod 2 = C((n+61)^6,n+63) mod 2 etc. - Paolo P. Lava, Aug 31 2007

Any sequence formed from zeros and a unique 1 can be produced using the formula a(n) = C(n^2k,n+2) mod 2, where k is a positive integer and n>=0. The sequence is formed by [2^2k-2 initial zeros] U [1] U [infinitely many zeros]. If we want to have 1 in a specific position the formula must be modified: a(n) = C((n+m)^2k,n+2+m) mod 2, where k and m are positive integers and n>=0. In this way we have {2^2k-2-m initial zeros} U {1} U {infinitely many zeros}. Of course we must have 2^2k-2>m. Then if we want the unique 1 in the position r, the minimum power k we can use is given by the relation 2^2k-1 >= r, namely k>=(1/2)*Log_2 (r+1). - Paolo P. Lava, Aug 31 2007

G.f.: x . E.g.f.: x . - Philippe Deléham, Nov 25 2008

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

MAPLE

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

MATHEMATICA

Table[If[n == 1, 1, 0], {n, 0, 104}] (* Robert G. Wilson v, Sep 30 2006 *)

PROG

(Haskell)

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

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

CROSSREFS

Cf. A000007, A008683, A000012, A007427, A000005, A007428, A007425, A227005, A227257, A227301, A003763, A209077.

Sequence in context: A157928 A159075 A178333 * A084928 A033683 A216284

Adjacent sequences:  A063521 A063522 A063523 * A063525 A063526 A063527

KEYWORD

easy,nonn,mult

AUTHOR

Labos Elemer, Jul 30 2001

STATUS

approved

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Last modified April 18 03:34 EDT 2014. Contains 240688 sequences.