$f(n) = 2f(n-1) -f(n-2) + 1$ find closed form by repeated substitution

Question

$$f(0)=a$$ $$f(1)=b$$

$$f(n) = 2f(n-1) -f(n-2) + 1$$ How can I begin repeated substitution with this? I'm confused because there are two $f$ terms not sure how to sub for both of them.

Answer 1

After some computations $$f(2) = 2f(1)-f(0)+1 = 2b-a+1,$$ $$f(3) = 2f(2)-f(1)+1 = 4b-2a+2-b+1 = 3b-2a+3,$$ $$f(4) = 2f(3)-f(2)+1 = 6b-4a+6-2b+a-1+1 = 4b-3a+6,$$ you can conjecture that $$f(n) = nb-(n-1)a+ \frac{n(n-1)}{2}$$ and prove this by induction.

Answer 2

I like to express recursions of this type by matrix-expressions, such that with a initial vector $[1,a,b]$ we get by the transfer matrix $M$ the vector $[1,b,c]$ where $b$ and $c$ are defined by one step of the recursion.
For your problem $$ M = \begin{bmatrix} 1&0&1 \\\ 0&0&-1 \\\ 0&1&2 \end{bmatrix}$$ and then $$ [1,a_1,a_2] \cdot M^h = [1,a_h,a_{h+1}] \tag 0 $$ where $h$ is then the iteration-"height".

Often diagonalization of the transfermatrix $M$ allow then the direct expression of the matrixpower $M^h$ in a formula with $h$ kept indeterminate.

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In your case I don't find a diagonalization for $M$ but a "similarity"-transformation (or "Schur-decomposition") of $M$ into $D$ such that with the orthogonal ("rotation"-) matrix $T$ and a triangular matrix $D$ $$ M = T^\tau \cdot D \cdot T \tag 1$$ such that $$ M^h = T^\tau \cdot D^h \cdot T \tag 2 $$
Here the superpostfix $\tau$ at matrixname $T$ means "transpose" but, because $T$ is constructed to be orthogonal, means also the inverse and thus we have indeed a "similarity" transformation.

Here $D$ has the identity-diagonal and $D^h = \exp(h \cdot \log(D))$ can be given in exact polynomial terms of $h$.
Let's denote $q = \sqrt {0.5}$ then I found that a valid solution for $T$ is $$ T= \begin{bmatrix} 0&0&1 \\\ q&-q&0 \\\ q&q&0 \end{bmatrix} \tag 3 $$ then
$$ D= T^\tau \cdot M \cdot T= \begin{bmatrix} 1&0&0 \\\ 2&1&0 \\\ q&q&1 \end{bmatrix} \tag 4$$ The matrix-logarithm of $D$ is then $$ \Lambda= \log(D)= \begin{bmatrix} 0&0&0 \\\ 2&0&0 \\\ 0&q&0 \end{bmatrix} \tag 5$$ and $D^h$ has then the simple form $$ D^h = \exp(h \cdot \Lambda) = \begin{bmatrix} 1&0&0 \\\ 2h&1&0 \\\ q \cdot h^2&q \cdot h&1 \end{bmatrix} \tag 6 $$ and $M^h$ is then $$M^h = T \cdot D^h \cdot T^\tau = \begin{bmatrix} 1& \frac12 h(h-1)& \frac 12 h(h+1) \\\ 0&1-h&-h \\\ 0& h&h+1 \end{bmatrix} \tag 7$$ and thus using eq $(0)$ $$ [1, a_1, a_2] \cdot M^h = [1, a_h, a_{h+1}] \\\ a_{h+1}= \left([1,a_1,a_2] \cdot M^h\right) [3] $$ we get $$ a_{h+1} = \frac12h(h+1) -h\cdot a_1 + (h+1) a_2 \tag 8$$

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Remarks:

• the equation (5) can exactly be determined by the mercatorseries for $(D-I)$ because $D-I$ is nilpotent and the series collapses to a finite sum.
• the equation (6) can exactly be determined by the common exponential-series on $ h\cdot \Lambda$ because $ h\cdot \Lambda$ is nilpotent and the exponential series collapses to a finite sum.
• In light of some other questions of the same type but where the triangular matrix D has entries $\ne 1$ in the diagonal it might be even better, more general and thus the "canonical" solution, to use the Jordan-decomposition $M=S \cdot J \cdot S^{-1}$ instead where the matrix J has an even simpler form than D and the matrix-log and -exponential can as easy be computed as with the diagonalization in the diagonalizable cases. Unfortunately, the Jordan-decomposition is tedious even with $3\times3$-matrices (but can for instance be called from Wolfram-Alpha) so I proposed the triangular decomposition-version here.

Answer 3

Since $$ f(n)-2f(n-1)+f(n-2)=1 $$ if we let $$ g(n)=f(n)-\frac{n^2}2 $$ we get $$ g(n)-2g(n-1)+g(n-2)=0 $$ the solution to which is $g(n)=cn+d$, since the characteristic polynomial is $(x-1)^2$. Thus, because $f(0)=a$ and $f(1)=b$, $$ f(n)=\frac{n^2-n}2+(b-a)n+a $$

Answer 4

A standard way to do this is through generating function.

The generating function of $f(n)$ is $$ g(z)=\sum_{n \ge 0} f(n)z^n $$ Multiply both sides of the recursion by $z^{n}$ gives $$ z^{n}f(n+2) = 2 z^{n} f(n+1) - z^{n}f(n) + z^n $$ Sum over $n \ge 0$ and we get $$ \frac{-a-b z+g(z)}{z^2}=\frac{2 (g(z)-a)}{z}-g(z)+\frac{1}{1-z} $$ Solving this gives $$ g(z)=\frac{-2 a z^2+3 a z-a+b z^2-b z-z^2}{(z-1)^3} $$ Now you can get the coefficient of $g(z)$ by applying a binomial expansion to $(z-1)^{-3}$.