A112299 - OEIS

%I #38 Sep 08 2022 08:45:21

%S 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,

%T 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,

%U -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

%N Expansion of x * (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^8) in powers of x.

%C Periodic with period length 8.

%C Sum_{k>=1} a(k)/k = Pi/8. - _Jaume Oliver Lafont_, Oct 20 2009

%H G. C. Greubel, <a href="/A112299/b112299.txt">Table of n, a(n) for n = 1..2500</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,-1,0,-1).

%F Euler transform of length 8 sequence [-1, -1, -1, 0, 0, 0, 0, 1].

%F Multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4).

%F G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x^2) * (1 + x^4)).

%F G.f.: f(x) - f(x^2) where f(x) := x / (1 + x^2). - _Michael Somos_, Jun 19 2015

%F a(n) = -a(4 - n) = a(n + 8) for all n in Z. a(4*n) = 0.

%F a(2*n) = - A056594(n-1). a(2*n + 1) = A033999(n). a(4*n + 1) = 1. a(4*n + 3) = -1. a(4*n + 2) = - A033999(n). - _Michael Somos_, Jun 19 2015

%F a(n) = A257196(n) unless n=0. - _Michael Somos_, Sep 01 2015

%e G.f. = x - x^2 - x^3 + x^5 + x^6 - x^7 + x^9 - x^10 - x^11 + x^13 + x^14 - x^15 + ...

%t LinearRecurrence[{0,-1,0,-1,0,-1},{1,-1,-1,0,1,1},110] (* _Harvey P. Dale_, Dec 07 2014 *)

%t a[ n_] := {1, -1, -1, 0, 1, 1, -1, 0}[[Mod[n, 8, 1]]];

%o (PARI) {a(n) = [0, 1, -1, -1, 0, 1, 1, -1][n%8 + 1]};

%o (PARI) {a(n) = [0, 1, -(-1)^(n\4), -1][n%4 + 1]}; /* _Michael Somos_, Jun 19 2015 */

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x+x^2)*(1-x)^2/((1+x^2)*(1+x^4)))); // _G. C. Greubel_, Aug 03 2018

%Y Cf. A033999, A056594, A257196.

%K sign,mult,easy

%O 1,1

%A _Michael Somos_, Sep 02 2005