A057427 - OEIS

%I #179 Jan 07 2023 12:39:24

%S 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,

%T 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,

%U 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

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

%C Parity of (n+1)-st prime, A000040(n+1). - _Philippe Deléham_, Apr 04 2009

%C Decimal expansion of 1/90.

%C Partial sums of A063524 (characteristic function of 1). - _Jeremy Gardiner_, Sep 08 2002

%C Characteristic function of positive integers. - _Franklin T. Adams-Watters_, Aug 01 2011

%C 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

%C Central terms of the triangle in A152487. - _Reinhard Zumkeller_, Dec 06 2008

%C 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.

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

%H Vincenzo Librandi, <a href="/A057427/b057427.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

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

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

%F a(A000027(n)) = 1; a(A000004(n)) = 0. - _Reinhard Zumkeller_, Oct 11 2008

%F a(n) = A000007(0^n). - _Jaume Oliver Lafont_, Mar 19 2009

%F From _Michael Somos_, Aug 17 2015: (Start)

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

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

%F Dirichlet g.f.: zeta(s) - 1. - _Álvar Ibeas_, Nov 29 2015; corrected by _Francois Oger_, Oct 26 2019

%F a(n) = A001065(n+1) - A048050(n+1). - _Omar E. Pol_, Apr 30 2018

%F E.g.f.: e^x - 1. - _Francois Oger_, Oct 26 2019

%F a(n) = 1-A000007(n). - _Chai Wah Wu_, Nov 14 2022

%e 1/90 = .0111111111111111111...

%e G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + ...

%p A057427:= signum: seq(A057427(k), k=0..50); # _Wesley Ivan Hurt_, Oct 22 2013

%t Table[Sign[n], {n, 0, 104}] (* _Arkadiusz Wesolowski_, Sep 16 2012 *)

%t CoefficientList[Series[x/((1 - x)), {x, 0, 25}], x]

%t LinearRecurrence[{1, 0}, {0, 1}, 105]

%t Array[Sign, 105, 0]

%t N[1/9, 111]

%t PadRight[{0},120,1] (* _Harvey P. Dale_, Jan 07 2023 *)

%o (PARI) {a(n) = sign(n)};

%o (PARI) /* n>=0 */ a(n)=!!n \\ _Jaume Oliver Lafont_, Mar 19 2009

%o (Haskell)

%o a057427 = signum

%o a057427_list = 0 : [1, 1 ..]  -- _Reinhard Zumkeller_, Nov 28 2012

%o (Python)

%o def A057427(n): return int(n!=0) # _Chai Wah Wu_, Nov 14 2022

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

%Y See also A261012.

%K nonn,easy,mult,nice,cons

%O 0,1

%A _Henry Bottomley_, Sep 05 2000

%E Entry edited at the suggestion of _Robert G. Wilson v_ by _N. J. A. Sloane_, Aug 16 2015