A164116 Expansion of (1 - x) * (1 - x^4) / (1 - x^5) in powers of x.

1, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1

Offset: 0

Author

Michael Somos, Aug 10 2009

Keywords

• sign
• easy

Comments

This sequence with a(0) replaced by 2 appears, together with three other sequences, in the formula 2*exp(2*Pi*n*I/5) = 2*T(n,x) + S(n-1,x)*sqrt(2+phi)*I, with x = (phi-1)/2 and I = sqrt(-1), where phi = (1+sqrt(5))/2 (golden section) after reduction of powers using phi^2 = phi+1. T and S are Chebyshev polynomials from A053120 and A049310. This results in 2*exp(2*Pi*n*I/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2+phi)*I, with A(n) = a(n+5), B(n) = A080891(n), C(n) = A156174(n+4) and D(n) = A010891(n+3) for n >= 0. - Wolfdieter Lang, Feb 26 2014

Examples

			G.f. = 1 - x - x^4 + 2*x^5 - x^6 - x^9 + 2*x^10 - x^11 - x^14 + 2*x^15 - x^16 + ...
exp(2*Pi*3*I/5) = (0 - phi) + (1 - phi)*sqrt(2+phi)*I, with phi = (1+sqrt(5))/2. - _Wolfdieter Lang_, Feb 26 2014
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (-1, -1, -1, -1).

Crossrefs

Cf. A164115, A164118.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x^4)/(1+x+x^2+x^3+x^4))); // G. C. Greubel, Sep 22 2018

Mathematica

CoefficientList[Series[(1-x)(1-x^4)/(1-x^5),{x,0,110}],x] (* Harvey P. Dale, Sep 25 2013 *)

PARI

{a(n) = -(n==0) + [2, -1, 0, 0, -1][n%5 + 1]};

Formula

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

a(n) = a(-n) for all n in Z. a(n+5) = a(n) unless n=0 or n=-5.

G.f.: (1 - x^4)/(1 + x + x^2 + x^3 + x^4).

a(n) = (-1)^n * A164118(n). Convolution inverse of A164115.

a(n) = 2*0^mod(n,5) - 0^n - mod(mod(n+2,5),2). - Wesley Ivan Hurt, Apr 28 2015