A046978 Numerators of Taylor series for exp(x)*sin(x).

0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac
• easy

Comments

Period 8: repeat [0, 1, 1, 1, 0, -1, -1, -1].

Lehmer sequence U_n for R=2, Q=1. - Artur Jasinski, Oct 06 2008

4*a(n+6) = period 8: repeat -4,-4,0,4,4,4,0,-4 = A189442(n+1) + A189442(n+5). - Paul Curtz, Jun 03 2011

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = 0. - Michael Somos, Nov 27 2019

Examples

			G.f. = x + x^2 + x^3 - x^5 - x^6 - x^7 + x^9 + x^10 + x^11 - x^13 - x^14 - ...
1*x + 1*x^2 + (1/3)*x^3 - (1/30)*x^5 - (1/90)*x^6 - (1/630)*x^7 + (1/22680)*x^9 + (1/113400)*x^10 + ...
		

References

• G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).
• C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Crossrefs

Cf. A046979.

Programs

Maple

A046978 := n -> `if`(n mod 4 = 0,0,(-1)^iquo(n,4)): # Peter Luschny, Aug 21 2011

Mathematica

a = -((1 + I)/Sqrt[2]) + Sqrt[2]; b = (1 + I)/Sqrt[2]; Table[ Round[(a^n - b^n)/(a - b)], {n, 0, 200}] (* Artur Jasinski, Oct 06 2008 *)
Table[Sign[Cos[Pi*(n-2)/4]],{n,0,100}] (* Wesley Ivan Hurt, Oct 10 2013 *)
LinearRecurrence[{0,0,0,-1},{0,1,1,1},120] (* or *) PadRight[{},120,{0,1,1,1,0,-1,-1,-1}] (* Harvey P. Dale, Mar 17 2017 *)

PARI

{a(n) = (n%4 > 0) * (-1)^(n\4)}; /* Michael Somos, Jul 16 2006 */

Formula

Euler transform of length 8 sequence [1, 0, -1, -1, 0, 0, 0, 1]. - Michael Somos, Jul 16 2006

G.f.: x * (1 + x + x^2) / (1 + x^4) = x * (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^8)). a(-n) = a(n + 4) = -a(n). - Michael Somos, Jul 16 2006

a(n) = round((b^n - c^n)/(b - c)) where b = sqrt(2)-((1+i)/sqrt(2)), c = (1+i)/sqrt(2). - Artur Jasinski, Oct 06 2008

a(n) = sign(cos(Pi*(n-2)/4)). - Wesley Ivan Hurt, Oct 02 2013