OFFSET
0,1
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
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.
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
EXAMPLE
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 + ...
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 *)
PROG
(PARI) {a(n) = (n%4 > 0) * (-1)^(n\4)}; /* Michael Somos, Jul 16 2006 */
CROSSREFS
KEYWORD
sign,frac,easy
AUTHOR
STATUS
approved