MATH SOLVE

2 months ago

Q:
# Find a compact form for generating function of the sequence 4, 4, 4, 4, 1, 0, 1, 0, 1, 0, 1, 0,

Accepted Solution

A:

The generating function for this sequence is[tex]f(x)=4+4x+4x^2+4x^3+x^4+x^6+x^8+\cdots[/tex]assuming the sequence itself is {4, 4, 4, 4, 1, 0, 1, 0, ...} and the 1-0 pattern repeats forever (as opposes to, say four 4s appearing after every four 1-0 pairs). We can make this simpler by "displacing" the odd-degree terms and considering instead the generating function,[tex]f(x)=3+4x+3x^2+4x^3+\underbrace{(1+x^2+x^4+x^6+x^8+\cdots)}_{g(x)}[/tex]where the coefficients of [tex]g(x)[/tex] follow a much more obvious pattern of alternating 1s and 0s. Let[tex]g(x)=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]where [tex]a_n[/tex] is recursively given by[tex]\begin{cases}a_0=1\\a_1=0\\a_{n+2}=a_n&\text{for }n\ge0\end{cases}[/tex]and explicitly by[tex]a_n=\dfrac{1+(-1)^n}2[/tex]so that[tex]g(x)=\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n[/tex]and so[tex]\boxed{f(x)=3+4x+3x^2+4x^3+\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n}[/tex]