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. - Wolfdieter Lang, Jan 23 2007
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
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..100000
G. P. Michon, Multiplicative Functions
Wikipedia, Dirichlet convolution
Index entries for linear recurrences with constant coefficients, signature (1).
FORMULA
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
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)
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 *)
LinearRecurrence[{1}, {0, 1, 0}, 106] (* Ray Chandler, Jul 15 2015 *)
PROG
(Haskell)
a063524 = fromEnum . (== 1) -- Reinhard Zumkeller, Apr 01 2012
(PARI) a(n)=n==1; \\ Charles R Greathouse IV, Apr 01 2012
(Python)
def A063524(n): return int(n==1) # Chai Wah Wu, Feb 04 2022
CROSSREFS
KEYWORD
easy,nonn,mult
AUTHOR
Labos Elemer, Jul 30 2001
STATUS
approved