I have encountered some polynomials in my work that have the following form:
$n=2$: $1$
$n=3$: $\phi^2 + 2\phi + 1$
$n=4$: $\phi^4 + 2\phi^3 + 4\phi^2 + 2\phi + 1$
$n=5$: $\phi^6 + 2\phi^5 + 4\phi^4 + 6\phi^3 + 4\phi^2 + 2\phi + 1$
$n=6$: $\phi^8 + 2\phi^7 + 4\phi^6 + 6\phi^5 + 9\phi^4 + 6\phi^3 + 4\phi^2 + 2\phi + 1$
$n=7$: $\phi^{10} + 2\phi^9 + 4\phi^8 + 6\phi^7 + 9\phi^6 + 12\phi^5 + 9\phi^4 + 6\phi^3 + 4\phi^2 + 2\phi + 1$
$n=8$: $\phi^{12} + 2\phi^{11} + 4\phi^{10} + 6\phi^9 + 9\phi^8 + 12\phi^7 + 16\phi^6 + 12\phi^5 + 9\phi^4 + 6\phi^3 + 4\phi^2 + 2\phi + 1$
$n=9$: $\phi^{14} + 2\phi^{13} + 4\phi^{12} + 6\phi^{11} + 9\phi^{10} + 12\phi^9 + 16\phi^8 + 20\phi^7 + 16\phi^6 + 12\phi^5 + 9\phi^4 + 6\phi^3 + 4\phi^2 + 2\phi + 1$
I am trying to figure out if there is a closed-form expression for the polynomial.
The coefficients follow the "quarter squares" sequence, which looks like this:
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| floor($\frac{n^2}{4}$) | 0 | 0 | 1 | 2 | 4 | 6 | 9 | 12 | 16 | 20 | 25 | 30 | 36 | 42 | 49 | 56 | 64 | 72 |
To avoid using the floor/ceiling function, the sequence can be defined also as "the square numbers and pronic numbers interleaved", with pronic number defined as "the product of two consecutive integers". Or as a piecewise function:
- $ q(n) = \frac{n^2}{4}$ if n is even
- $ q(n) = \frac{n-1}{2} \cdot \frac{n+1}{2}$ if n is odd
Using sums and using $q(n)$ to denote the $n$-th element of the sequence above, we get the following form:
$q(n) \cdot \phi^{n - 2} + \sum_{i=1}^{n-2} \left(q(n-i) \cdot \left(\phi^{n - 2 + i} + \phi^{n - 2 - i}\right) \right)$
Does anyone have an idea of how to simplify these polynomials?
