The stamp folding problem asks for the number of ways to fold a strip of stamps, with variants determined by which features of a folding are distinguished. Considering
only positions of the hinges for unlabeled stamps without regard to orientation of
the stamps, the number of foldings is denoted 
. If the stamps are labeled and orientation is taken into
account, the number of foldings is denoted 
. Finally, the number of symmetric foldings is denoted 
. The following table summarizes these
values for the first 
.
 |  |  |  |
| Sloane | A001010 | A001011 | A000136 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 1 | 2 |
| 3 | 2 | 2 | 6 |
| 4 | 4 | 5 | 16 |
| 5 | 6 | 14 | 50 |
| 6 | 8 | 38 | 144 |
| 7 | 18 | 120 | 462 |
| 8 | 20 | 353 | 1392 |
| 9 | 56 | 1148 | 4536 |
| 10 | 48 | 3527 | 14060 |
For labeled stamp foldings, Sakai (2026) described a boundary-based decomposition in which the second stamp is fixed as a boundary. Under suitable conditions, the portions above and below this boundary can be treated independently, giving a product structure that explains the appearance of recurrence relations. The decomposition is structural rather than a closed counting formula, and it does not extend directly to variants with a fixed boundary or additional symmetries.
