A057427 a(n) = 1 if n > 0, a(n) = 0 if n = 0; series expansion of x/(1-x).

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

Offset: 0

Author

Henry Bottomley, Sep 05 2000

Keywords

• nonn
• easy
• mult
• nice
• cons

Comments

Parity of (n+1)-st prime, A000040(n+1). - Philippe Deléham, Apr 04 2009

Decimal expansion of 1/90.

Partial sums of A063524 (characteristic function of 1). - Jeremy Gardiner, Sep 08 2002

Characteristic function of positive integers. - Franklin T. Adams-Watters, Aug 01 2011

Number of binary bracelets of n beads, 0 of them 0. Number of binary bracelets of n beads, 1 of them 0. Number of binary bracelets of n beads, 0 of them 0, with 00 prohibited. For n>=2, a(n-1) is the number of binary bracelets of n beads, one of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008

Central terms of the triangle in A152487. - Reinhard Zumkeller, Dec 06 2008

This is sgn(n) (or sign(n), or signum(n)) restricted to nonnegative integers. See sequence A261012 for a version that extends the sequence backwards to offset -1.

Examples

			1/90 = .0111111111111111111...
G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + ...
		

References

• T. M. MacRobert, Functions of a Complex Variable, 4th ed., Macmillan and Co., London, 1958, p. 90.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for characteristic functions
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A000004, A000007, A000012, A000027, A000040, A063524, A152487.

See also A261012.

Programs

Haskell

a057427 = signum
a057427_list = 0 : [1, 1 ..]  -- Reinhard Zumkeller, Nov 28 2012

Maple

A057427:= signum: seq(A057427(k), k=0..50); # Wesley Ivan Hurt, Oct 22 2013

Mathematica

Table[Sign[n], {n, 0, 104}] (* Arkadiusz Wesolowski, Sep 16 2012 *)
CoefficientList[Series[x/((1 - x)), {x, 0, 25}], x]
LinearRecurrence[{1, 0}, {0, 1}, 105]
Array[Sign, 105, 0]
N[1/9, 111]
PadRight[{0},120,1] (* Harvey P. Dale, Jan 07 2023 *)

PARI

{a(n) = sign(n)};

PARI

/* n>=0 */ a(n)=!!n \\ Jaume Oliver Lafont, Mar 19 2009

Python

def A057427(n): return int(n!=0) # Chai Wah Wu, Nov 14 2022

Formula

G.f.: x / (1 - x).

G.f.: Sum_{k>=0} 2^k * x^(2^k) / (1 + x^(2^k)). - Michael Somos, Sep 11 2005

a(A000027(n)) = 1; a(A000004(n)) = 0. - Reinhard Zumkeller, Oct 11 2008

a(n) = A000007(0^n). - Jaume Oliver Lafont, Mar 19 2009

From Michael Somos, Aug 17 2015: (Start)

a(n) = -a(-n) for all n in Z if a(n) is treated as sgn(n).

Sum_{k<0} a(k) * x^k = 1 / (1 - x) if abs(x) > 1. (End)

Dirichlet g.f.: zeta(s) - 1. - Álvar Ibeas, Nov 29 2015; corrected by Francois Oger, Oct 26 2019

a(n) = A001065(n+1) - A048050(n+1). - Omar E. Pol, Apr 30 2018

E.g.f.: e^x - 1. - Francois Oger, Oct 26 2019

a(n) = 1-A000007(n). - Chai Wah Wu, Nov 14 2022

Extensions

Entry edited at the suggestion of Robert G. Wilson v by N. J. A. Sloane, Aug 16 2015