A111913 -id:A111913 - cp's OEIS frontend

A111912 Expansion of x(2 +3x +x^2 -2x^5 -x^7 -x^8)/((1-x)(1+x)*(1-x^4+x^8)).

0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1, 0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1, 0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1

Offset: 0

Creighton Dement, Aug 20 2005

Sequence has period 24.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1,0,-1,0,1).

Cf. A085846, A111913, A111914, A111915.

R:=PowerSeriesRing(Integers(), 75);
Coefficients(R!( x*(2+3*x+x^2-2*x^5-x^7-x^8)/((1-x)*(1+x)*(1-x^4+x^8)) )); // G. C. Greubel, Feb 12 2021

seq(coeff(series((x*(-2-3*x-x^2+2*x^5+x^7+x^8)/((x-1)*(x+1)*(x^8-x^4+1))),x,n+1),x,n),n=0..75); # Muniru A Asiru, Jun 06 2018

LinearRecurrence[{0,1,0,1,0,-1,0,-1,0,1}, {0,2,3,3,3,5,4,6,3,5}, 75] (* G. C. Greubel, Feb 12 2021 *)

Vec(x*(-2-3*x-x^2+2*x^5+x^7+x^8)/((x-1)*(x+1)*(x^8-x^4+1))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

def A111912_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(2+3*x+x^2-2*x^5-x^7-x^8)/((1-x)*(1+x)*(1-x^4+x^8)) ).list()
A111912_list(75) # G. C. Greubel, Feb 12 2021

A111914 Expansion of -x^2(x^4-2x^3+x^2-2x+1)(x+1)^2 / ((x-1)*(x^8-x^4+1)).

Original entry on oeis.org

0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4

Offset: 0

Creighton Dement, Aug 20 2005

It appears that (a(n)) has period 24.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,0,0,-1,1).

Cf. A111912, A111913, A111915, A085846.

concat([0,0], Vec(x^2*(1 + x)^2*(1 - 2*x + x^2 - 2*x^3 + x^4) / ((1 - x)*(1 - x^4 + x^8)) + O(x^40))) \\ Colin Barker, May 18 2019

a(n) = a(n-1) + a(n-4) - a(n-5) - a(n-8) + a(n-9) for n > 8. - Colin Barker, May 18 2019

A111915 Expansion of -x^2(x-1)(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

Original entry on oeis.org

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

Offset: 0

Creighton Dement, Aug 20 2005

It appears that a(n) has period 24.

This is true, as (1-x^4+x^8) is the cyclotomic polynomial for n=24. - Joerg Arndt, Feb 03 2017

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-1).

Cf. A085846, A111912, A111913, A111914.

CoefficientList[Series[-x^2*(x - 1)*(x^2 - x + 1)*(x + x^2 + 1)/(1 - x^4 + x^8), {x, 0, 100}], x] (* Wesley Ivan Hurt, Feb 03 2017 *)

a(n)=[0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1][n%24+1] \\ Charles R Greathouse IV, Feb 03 2017

G.f.: -x^2*(x-1)*(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

cp's OEIS Frontend

A111912 Expansion of x(2 +3x +x^2 -2x^5 -x^7 -x^8)/((1-x)(1+x)*(1-x^4+x^8)).

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A111914 Expansion of -x^2(x^4-2x^3+x^2-2x+1)(x+1)^2 / ((x-1)*(x^8-x^4+1)).

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A111915 Expansion of -x^2(x-1)(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

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cp's OEIS Frontend

A111912 Expansion of x*(2 +3*x +x^2 -2*x^5 -x^7 -x^8)/((1-x)*(1+x)*(1-x^4+x^8)).

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A111914 Expansion of -x^2*(x^4-2*x^3+x^2-2*x+1)*(x+1)^2 / ((x-1)*(x^8-x^4+1)).

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A111915 Expansion of -x^2*(x-1)*(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

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A111912 Expansion of x(2 +3x +x^2 -2x^5 -x^7 -x^8)/((1-x)(1+x)*(1-x^4+x^8)).

A111914 Expansion of -x^2(x^4-2x^3+x^2-2x+1)(x+1)^2 / ((x-1)*(x^8-x^4+1)).

A111915 Expansion of -x^2(x-1)(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).