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%I #44 Mar 07 2020 07:53:42
%S 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,
%T -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,
%U 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
%N Numerators of Taylor series for exp(x)*sin(x).
%C Period 8: repeat [0, 1, 1, 1, 0, -1, -1, -1].
%C Lehmer sequence U_n for R=2, Q=1. - _Artur Jasinski_, Oct 06 2008
%C 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
%C 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
%D G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).
%H C. Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%F Euler transform of length 8 sequence [1, 0, -1, -1, 0, 0, 0, 1]. - _Michael Somos_, Jul 16 2006
%F 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
%F 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
%F a(n) = sign(cos(Pi*(n-2)/4)). - _Wesley Ivan Hurt_, Oct 02 2013
%e G.f. = x + x^2 + x^3 - x^5 - x^6 - x^7 + x^9 + x^10 + x^11 - x^13 - x^14 - ...
%e 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 + ...
%p A046978 := n -> `if`(n mod 4 = 0,0,(-1)^iquo(n,4)): # _Peter Luschny_, Aug 21 2011
%t 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 *)
%t Table[Sign[Cos[Pi*(n-2)/4]],{n,0,100}] (* _Wesley Ivan Hurt_, Oct 10 2013 *)
%t 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 *)
%o (PARI) {a(n) = (n%4 > 0) * (-1)^(n\4)}; /* _Michael Somos_, Jul 16 2006 */
%Y Cf. A046979.
%K sign,frac,easy
%O 0,1
%A _N. J. A. Sloane_
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