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

%I A046978 #46 Jan 05 2025 19:51:35
%S A046978 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 A046978 -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 A046978 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 A046978 Numerators of Taylor series for exp(x)*sin(x).
%C A046978 Period 8: repeat [0, 1, 1, 1, 0, -1, -1, -1].
%C A046978 Lehmer sequence U_n for R=2, Q=1. - _Artur Jasinski_, Oct 06 2008
%C A046978 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 A046978 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 A046978 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%H A046978 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).
%H A046978 C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%F A046978 Euler transform of length 8 sequence [1, 0, -1, -1, 0, 0, 0, 1]. - _Michael Somos_, Jul 16 2006
%F A046978 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 A046978 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 A046978 a(n) = sign(cos(Pi*(n-2)/4)). - _Wesley Ivan Hurt_, Oct 02 2013
%e A046978 G.f. = x + x^2 + x^3 - x^5 - x^6 - x^7 + x^9 + x^10 + x^11 - x^13 - x^14 - ...
%e A046978 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 A046978 := n -> `if`(n mod 4 = 0,0,(-1)^iquo(n,4)): # _Peter Luschny_, Aug 21 2011
%t A046978 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 A046978 Table[Sign[Cos[Pi*(n-2)/4]],{n,0,100}] (* _Wesley Ivan Hurt_, Oct 10 2013 *)
%t A046978 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 A046978 (PARI) {a(n) = (n%4 > 0) * (-1)^(n\4)}; /* _Michael Somos_, Jul 16 2006 */
%Y A046978 Cf. A046979.
%K A046978 sign,frac,easy
%O A046978 0,1
%A A046978 _N. J. A. Sloane_