Every day is a holiday revisited

On planet J, a year lasts for $D$ days. Holidays are defined by the two following rules.
1. At the beginning of the reign of the current Emperor, his birthday is declared a holiday from that year onwards.
2. If both the day before and after a day $d$ are holidays, then $d$ also becomes a holiday.

Initially there are no holidays. Let $f(n,d)$ be the number of methods such that all the days of the year are holidays after exactly $d$ days, with the year lasts for $n$ days.

You are given $f(4,2)=4$, $f(10,5)=240$, $f(50,100)mod1000000007=263154905$.

Find $f(1000,{10}^{12})mod1000000007$.

---

Thanks to baihacker for the idea.

This problem was inspired by Problem 645 at Project Euler (https://projecteuler.net/problem=645)