A078020 Expansion of (1-x)/(1-x+2*x^2).

1, 0, -2, -2, 2, 6, 2, -10, -14, 6, 34, 22, -46, -90, 2, 182, 178, -186, -542, -170, 914, 1254, -574, -3082, -1934, 4230, 8098, -362, -16558, -15834, 17282, 48950, 14386, -83514, -112286, 54742, 279314, 169830, -388798, -728458, 49138, 1506054, 1407778, -1604330, -4419886, -1211226, 7628546

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Comments

Equals the INVERT transform of [1, -1, -1, 1, 1, -1, -1, 1, 1, ...], i.e., 1 followed by repeats of (-1, -1, 1, 1, ...). - Gary W. Adamson, Sep 16 2008

Pisano period lengths: 1, 1, 8, 1, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-2).

Crossrefs

Cf. A001607, A107920.

Programs

GAP

a:=[1,0];; for n in [2..50] do a[n]:=a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Jun 29 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x+2*x^2) )); // G. C. Greubel, Jun 29 2019

Mathematica

LinearRecurrence[{1,-2}, {1,0}, 50] (* or *) CoefficientList[Series[(1 - x)/(1-x+2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 29 2019 *)

PARI

Vec((1-x)/(1-x+2*x^2)+O(x^50)) \\ Charles R Greathouse IV, Sep 25 2012

Sage

((1-x)/(1-x+2*x^2)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

Formula

a(n) = A107920(n+1) - A107920(n). - R. J. Mathar, Mar 14 2011

a(n) = (-1)^n*(A001607(n) + A001607(n-1)). - G. C. Greubel, Jun 29 2019