A371606 Number of ways to fold with complete turns a strip of n blank double-sided sticky stamps.

1, 1, 2, 5, 14, 38, 116, 337, 1024, 3022, 9068, 26736, 79165, 231933, 679344, 1976199, 5738101

Offset: 1

Author

Klemen Klanjscek, Mar 29 2024

Keywords

• nonn
• more

Comments

The unlabeled sticky stamps have glue on both sides. Once two stamps are glued they cannot be separated. When constructing a folding we are not allowed to make partial folds (turns less than 180 degrees).

First 6 terms agree with the sequence A001011, afterwards a(n) < A001011(n).

Examples

			For n = 7 foldings (1 6 5 4 3 2 7), (4 5 6 1 7 2 3), (3 4 5 6 1 7 2), and (1 7 2 3 4 5 6) cannot be produced if stamps are sticky on both sides and we are only allowed to do complete folds. If stamps are not sticky and we are still only allowed to do complete folds, these foldings are still possible. For example, folding strategy for (1 6 5 4 3 2 7):
Unfolded:
 <1--2--3--4--5--6--7>
Step 1:
 /-3--4--5--6--7>
 \-2--1>
 Step 2:
 <7--6--5--4-\
         /-3-/
         \-2--1>
Step 3:
 <7--6-\
   /-5-/
   \-4-\
   /-3-/
   \-2--1>
Step 4:
 /---6-\
 | /-5-/
 | \-4-\
 | /-3-/
 | \-2--1>
 \---7>
Step 5:
    <1---\
 /---6-\ |
 | /-5-/ |
 | \-4-\ |
 | /-3-/ |
 | \-2---/
 \---7>
If stamps are sticky, this strategy fails, because after the first step stamps 1 and 4 cannot be separated (every other strategy also fails).
		

Links

• Klemen Klanjscek, Python program for counting sticky unlabeled stamp foldings

Crossrefs

Cf. A001011.

Programs

Python

# See Link

Extensions

a(15)-a(17) from Klemen Klanjscek, Jul 09 2024