The four sequences¶
All known terms of the four sequences, with the evidence behind each. Intervals mark terms that are not yet known exactly. All sequences are indexed by the number of lines \(n \ge 1\). Labels: proved, computer-verified, bound.
| \(n\) | \(a(n)\) arithmetic | \(b(n)\) combinatorial | \(c(n)\) realizable | \(d(n)\) values of \(f_0\) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 |
| 3 | 2 | 2 | 2 | 2 |
| 4 | 4 | 3 | 3 | 3 |
| 5 | 7 | 5 | 5 | 5 |
| 6 | 15 | 9 | 9 | 9 |
| 7 | 32 | 17 | 16 | 13 |
| 8 | 66 | 29 | 28 | 18 |
| 9 | 141 | 56 | 53 | 24 |
| 10 | 295 | 106 | 98 | 30 |
| 11 | 619 | 202 | 180 | 38 |
| 12 | 1286 | 384 | 336 | 46 |
How each column is established:
- \(a(n)\) is a count of partitions and is proved for every \(n\) by the identity with A072964 (details).
- \(b(n)\) is computer-verified: an exhaustive search over linear spaces decides every admissible type, with no time-outs, and a SAT solver re-derives the same answers (details).
- \(c(n)\) for \(n \le 12\) is computer-verified: every realizable type has an exact certificate checked by the verifier, every other type is excluded by a theorem or by the exhaustive orientability search (details). For \(n = 13\): 620 certificates and 39 undecided types.
- \(d(n)\) combines certificates (a value is attained) with theorems (a value is impossible); for \(n \le 13\) every value is decided (details). The term \(d(12) = 46\) uses the non-realizability of the three types with \(21\) points that pass both the combinatorial test and Melchior's inequality: \(\{4^9\}\) by a theorem, \(\{5^2, 4^3, 3^6\}\) and \(\{5^3, 3^9\}\) by the orientability search.
What is not in the OEIS¶
The arithmetic sequence is A072964 shifted. The other three were searched for in the OEIS (full sequences, subsequences of five consecutive terms, shifts): no entry matches. Two entries coincide with ours on a few terms by accident and must not be confused with them: A129230 (\(0, 1, 3, 5, 9, 13, 18, 24, 30, 38, 46, 55, \dots\), a golden-ratio floor sum) is shifted by two: its terms \(3, 5, \dots, 46\) coincide with \(d(4), \dots, d(12)\), and the next term is \(55 \ne d(13) = 56\); A005834 (\(0, 2, 2, 5, 8, 16, 28, 53, 98, 183, \dots\)) contains \(16, 28, 53, 98\) and then \(183 \ne 180\).