A244895 - OEIS

%I #22 Jun 25 2022 00:04:14

%S 0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,

%T -1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,

%U -1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1,1,-1,-1,0,1

%N Period 5: repeat [0, 1, 1, -1, -1].

%C This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = z = -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.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, -1, -1).

%F G.f.: x * (1 + x) * (1 - x^2) / (1 - x^5).

%F Euler transform of length 5 sequence [ 1, -2, 0, 0, 1].

%F a(n) = -a(-n) = a(n + 5) for all n in Z.

%F 0 = (a(n) + a(n+2)) * (a(n) - a(n+1) + a(n+2)) for all n in Z.

%F 0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z.

%F 0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.

%F |A011558(n)| = |A080891(n)| = |A100047(n)| = |a(n)|. - _Michael Somos_, May 24 2015

%F a(5*n) = 0, a(5*n + 1) = a(5*n + 2) = 1, a(5*n + 3) = a(5*n + 4) = -1 for all n in Z. -_Michael Somos_, Nov 27 2019

%e G.f. = x + x^2 - x^3 - x^4 + x^6 + x^7 - x^8 - x^9 + x^11 + x^12 + ...

%t a[ n_] := {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* _Michael Somos_, Jan 08 2015 *)

%t a[ n_] := Sign[ Mod[ n, 5, -2]]; (* _Michael Somos_, Jan 08 2015 *)

%t PadRight[{},120,{0,1,1,-1,-1}] (* _Harvey P. Dale_, Nov 11 2020 *)

%o (PARI) {a(n) = [0, 1, 1, -1, -1][n%5 + 1]};

%o (PARI) {a(n) = sign( centerlift( Mod(n, 5)))};

%Y Cf. A011558, A080891, A100047.

%K sign,easy

%O 0,1

%A _Michael Somos_, Jul 07 2014