A154272 1,0,1 followed by 0,0,0,...

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

Offset: 1

Author

Mats Granvik, Jan 06 2009

Keywords

• nonn
• mult
• easy

Comments

Dirichlet inverse of this sequence is A154271. There is progression of sequences starting with A000007, A019590 and then this sequence A154272. From A019590 onwards the Dirichlet inverse of such sequences appears to be positive as often as negative. Except for the first term, the all 1's sequence A000012, is a union of the 1's in sequences A000007, A019590, A154272 etc. It therefore in a sense seems likely that the Moebius function is positive as often as negative because the Dirichlet inverse of A000012 is the Moebius function A008683. However the whole is more than the sum of its parts and the likelihood of the signs of the Moebius function cannot be inferred.

With offset -1, this is the replicable function number 1a. The generating function A(x) = 1/x + x is the first modular "fiction". This is a completely 2-replicable function. The g.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v + 2 - u^2. Similarly the replicable function number 2b gives the sequence 1,0,-1,0,0,0,0,0,... - Michael Somos, Aug 04 2009

The g.f. x+x^3 equals x*Phi(4,x), where Phi are the cyclotomic polynomials. The series reversion of y=x+x^3 is x = y - y^3 + 3*y^5 - 12*y^7 + 55*y^9 - ..., which is a signed variant of A001764. - R. J. Mathar, Sep 29 2012

References

• I. Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963.

Links

• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
• J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
• Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A154271, A154269, A000012, A000007, A019590, A008683.

Programs

Mathematica

PadRight[{1,0,1},150,0] (* Harvey P. Dale, Jun 14 2017 *)

PARI

{a(n) = (n==1) || (n==3)}; /* Michael Somos, Jan 03 2013 */

Formula

a(n)=1 if m(n) = 1/sin(Pi/(2*n)) is a natural number, and 0 otherwise. m(1)=1 and m(3)=2. See the quoted I. Niven book, Corollary 3.12, p.41. - Wolfdieter Lang, Dec 17 2010

Dirichlet g.f. 1+1/3^s. - R. J. Mathar, Mar 12 2012

Euler transform of length 4 sequence [ 0, 1, 0, -1]. - Michael Somos, Aug 04 2009

G.f.: x + x^3 = x / (1 - x^2 / (1 + x^2)) = x * (1 - x^4) / (1 - x^2). - Michael Somos, Jan 03 2013

a(n) = Sum_{d|n} mu(n/d) * tau(gcd(d,3)). - Ridouane Oudra, Apr 28 2025

a(n) = (Sum_{d|n} mu(n/d) * gcd(d,3)) / phi(n). - Ridouane Oudra, Apr 30 2025