2016-05-18 Wechselgeld.sagews

Wechselgeld

R.<z> = PowerSeriesRing(ZZ)

1 / ((1-z) * (1-z^2) * (1-z^5))

1 + z + 2*z^2 + 2*z^3 + 3*z^4 + 4*z^5 + 5*z^6 + 6*z^7 + 7*z^8 + 8*z^9 + 10*z^10 + 11*z^11 + 13*z^12 + 14*z^13 + 16*z^14 + 18*z^15 + 20*z^16 + 22*z^17 + 24*z^18 + 26*z^19 + O(z^20)

Partialbruchzerlegung

var('z')

z

F = 1 / ((1-z) * (1-z^2) * (1-z^5))
F

-1/((z^5 - 1)*(z^2 - 1)*(z - 1))

show(F.partial_fraction())

$\displaystyle \frac{z^{3} + 2 \, z^{2} + z + 1}{5 \, {\left(z^{4} + z^{3} + z^{2} + z + 1\right)}} + \frac{1}{8 \, {\left(z + 1\right)}} - \frac{13}{40 \, {\left(z - 1\right)}} + \frac{1}{4 \, {\left(z - 1\right)}^{2}} - \frac{1}{10 \, {\left(z - 1\right)}^{3}}$