OFFSET
0,2
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.
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, C# program for A342759
N. J. A. Sloane, Illustration of G(n) for n = 0..4
EXAMPLE
See illustration in Links section.
PROG
(C#) See Links section.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Rémy Sigrist and N. J. A. Sloane, Mar 21 2021
STATUS
approved