cp's OEIS Frontend

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.

%I A342759 #29 May 02 2021 10:47:10
%S A342759 1,2,3,4,6,10,16,25,43,73,133,241,457,865,1681,3265,6433,12673,25153,
%T A342759 49921,99457,198145,395521,789505,1577473,3151873,6300673,12595201,
%U A342759 25184257,50356225,100700161
%N 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.
%C A342759 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.
%C A342759 After two folds, we again have a (smaller and thicker) square, and we repeat the process.
%C A342759 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.
%C A342759 The number of vertices of degree 1 after n+1 folds appears to be A274230(n).
%C A342759 We ignore the folk theorem that says no sheet of paper can be folded more than seven times.
%D A342759 Rémy Sigrist and N. J. A. Sloane, Notes on Two-Dimensional Paper-Folding, Manuscript in preparation, April 2021.
%H A342759 J.-P. Allouche and M. Mendes France, <a href="https://webusers.imj-prg.fr/~jean-paul.allouche/allmendeshouches.pdf">Automata and Automatic Sequences</a>, 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.
%H A342759 J.-P. Allouche and M. Mendes France, <a href="/A003842/a003842.pdf">Automata and Automatic Sequences</a>, 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]
%H A342759 Rémy Sigrist, <a href="/A342759/a342759.png">Illustration of initial terms</a>
%H A342759 Rémy Sigrist, <a href="/A342759/a342759_1.png">Illustration of the number of vertices of degree 1 for n = 0..8</a>
%H A342759 Rémy Sigrist, <a href="/A342759/a342759.txt">C# program for A342759</a>
%H A342759 N. J. A. Sloane, <a href="/A342759/a342759.pdf">Illustration of G(n) for n = 0..4</a>
%e A342759 See illustration in Links section.
%o A342759 (C#) See Links section.
%Y A342759 Cf. A001511, A014577, A146528, A274230, A342760, A342761, A342762, A342763, A342764.
%K A342759 nonn,more
%O A342759 0,2
%A A342759 _Rémy Sigrist_ and _N. J. A. Sloane_, Mar 21 2021