A342759 Fold a square sheet of paper alternately vertically to the left and horizontally downwards; after each fold, draw a line along each inward crease; after n folds, the resulting graph has a(n) regions.

1, 2, 3, 4, 6, 10, 16, 25, 43, 73, 133, 241, 457, 865, 1681, 3265, 6433, 12673, 25153, 49921, 99457, 198145, 395521, 789505, 1577473, 3151873, 6300673, 12595201, 25184257, 50356225, 100700161

Offset: 0

Author

Rémy Sigrist and N. J. A. Sloane, Mar 21 2021

Keywords

• nonn
• more

Comments

Take a square sheet of paper and fold it first vertically and then horizontally so that the bottom right corner stays in place. After each fold, unfold the paper and draw a line along each crease that is indented inwards (along which water would flow); upward creases (ridges) are not marked.

After two folds, we again have a (smaller and thicker) square, and we repeat the process.

After n individual folds, when the paper is unfolded the lines form a planar graph G(n). The numbers of regions, vertices, edges, and connected components in G(n) are given in the present sequence, A146528 (still to be confirmed), A342761, and A342762.

The number of vertices of degree 1 after n+1 folds appears to be A274230(n).

We ignore the folk theorem that says no sheet of paper can be folded more than seven times.

Examples

			See illustration in Links section.
		

References

• Rémy Sigrist and N. J. A. Sloane, Notes on Two-Dimensional Paper-Folding, Manuscript in preparation, April 2021.

Links

• J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
• J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
• Rémy Sigrist, Illustration of initial terms
• Rémy Sigrist, Illustration of the number of vertices of degree 1 for n = 0..8
• Rémy Sigrist, C# program for A342759
• N. J. A. Sloane, Illustration of G(n) for n = 0..4

Crossrefs

Cf. A001511, A014577, A146528, A274230, A342760, A342761, A342762, A342763, A342764.