A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

-1, 1, 2, 5, 4, 1, -8, -15, -16, 1, 32, 65, 64, 1, -128, -255, -256, 1, 512, 1025, 1024, 1, -2048, -4095, -4096, 1, 8192, 16385, 16384, 1, -32768, -65535, -65536, 1, 131072, 262145, 262144, 1, -524288, -1048575, -1048576, 1, 2097152, 4194305, 4194304, 1, -8388608, -16777215, -16777216, 1, 33554432

Offset: 0

Author

Creighton Dement, May 13 2005

Keywords

• easy
• sign

Comments

Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

Links

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

Crossrefs

Cf. A000035, A010673, A078069, A090131, A099087, A146559.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // G. C. Greubel, Sep 08 2021

Mathematica

CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* Harvey P. Dale, Mar 20 2013 *)

SageMath

def A106664_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
A106664_list(50) # G. C. Greubel, Sep 08 2021

Formula

a(n) = (1/2)*(A010673(n) - A099087(n+2)).

a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - Ralf Stephan, Nov 16 2010

From G. C. Greubel, Sep 08 2021: (Start)

a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).

a(n) = A000035(n) - A146559(n).

E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)