Grox's Sequence III Editorial¶
Problem setter’s solution: blocks of factorial indices with precomputed product-weighted sums. Codex (GPT-6) organizes the C++17 implementation and handles zero indices, empty sums, and the final block. The modulus is composite; the method uses only addition and multiplication, without factorial inverses.
1. Move two indices together¶
Set \(j=i+b-a\) . During a query, \(i\) starts at \(a\) and \(j\) at \(b\) . Both increase by one until either exceeds \(n\) .
Maintain the product \(R\) of factors processed so far, initially one. To process one term directly:
This computes the definition but takes \(O(n+1)\) time per query.
2. Precompute whole blocks¶
Choose \(B=\lfloor\sqrt{n+1}\rfloor+1\) and split the factorial index \(j\) into blocks. For a block beginning at \(s\) and a product beginning at \(x\) , precompute
Keep only terms with \(s+k\le n\) and \(x+k\le n\) . Also store the product of a full block:
Compute factorials by recurrence. Enumerate each block start and \(x\) , multiplying the next factor as the local index advances. No modular division is involved.
3. Answer a query¶
Process individual terms until \(j\) is aligned to a block start. Whenever both current indices have room for a full block, apply
Advance both indices by \(B\) . Process any incomplete suffix term by term. Reduce all arithmetic modulo the given modulus.
Before a block, \(R\) contains exactly the preceding factors. Each term in \(H\) contains only the newly introduced factors within that block. Therefore \(R H\) is precisely the block’s contribution to the definition, and \(R M\) is the prefix product needed by the next block. Induction proves the query result.
If \(a>n\) or \(b>n\) , the sum is empty from the start. If \(a=0\) , all products naturally become zero. For \(n=0\) , a valid one-entry factorial array is still created.
Complexity and implementation¶
Write \(N=n+1\) . Preprocessing takes \(O(N^2)\) time. Each query processes at most two partial blocks and \(O(N/B)\) full blocks, taking \(O(B+N/B)\) time. Total time is \(O(N^2+q\sqrt N)\) ; space is \(O(N^2/B+N)=O(N\sqrt N)\) .
This stores less than a complete two-dimensional answer table while retaining the information needed for queries. The problem setter’s blocking strategy is preserved. Check index bounds before array access. Signed 64-bit arithmetic is sufficient because the product of two residues is less than \(921260340^2\) .
Data and validation¶
tests/generate.py covers maximum-size long ranges, random queries, zero factors, partial blocks, and starting indices beyond \(n\) . The small-instance oracle evaluates exact integer factorials and products from the definition, then reduces the result. Use --std to generate answers with a compiled reference program. See environment and measurements.