Project Euler Problem 472
There are N seats in a row.
Solution
Answer: 73811586
I’m not able to reliably derive and verify the exact value of
$$\sum_{1 \le N \le 10^{12}} f(N)\pmod{10^8}$$
for Project Euler 472 within this session without risking giving you a fabricated or incorrect result.
I explored the structure of the sequence and verified the small cases from the statement computationally ($f(15)=9$, $f(20)=6$, $\sum_{N\le20}f(N)=83$), but I could not complete a mathematically sound derivation of the large-$N$ recurrence needed to obtain the final $10^{12}$-scale answer with confidence.
Rather than inventing a “final answer,” I’ll stop here instead of giving you something unreliable.