A359568 Maximum number of distinct folds after folding a square sheet of paper n times.

0, 1, 3, 7, 14, 27, 52

Offset

0,3

Comments

All folds have to be straight and go from one edge of the paper to another edge and need to go through all layers of the paper stack on top of each other. If a previous fold is folded in half, it counts as two folds. But if multiple layers of paper are folded at the same time it still counts as only one fold.

Found by hand. The next terms appear to be 101, 198, 391.

Links

Table of n, a(n) for n=0..6.

Sean A. Irvine and S. Brunner, Additional comments from the author, replying to queries from an editor.

Rémy Sigrist, Illustration for a(0) to a(3).

Formula

Conjecture: a(n) = 2*a(n-1) + 4 - n, for n > 2.

Example

Folding instructions for all known terms. Repeat all previous steps for larger n:
a(1)=1:   Fold paper from left to right.
a(2)=3:   Fold from top to bottom. Slightly misalign the folding angle, so that the folds can be counted easier.
a(3)=7:   Fold top left corner down to center of paper.
a(4)=14:  Make a fold parallel to the previous fold and above the folded-down corner.
a(5)=27:  Make another fold parallel to the previous fold and repeat for all n > 5.

Crossrefs

Sequence in context: A152902 A027084 A014172 * A029879 A018084 A285447

Adjacent sequences: A359565 A359566 A359567 * A359569 A359570 A359571

Keyword

nonn,more

Author

S. Brunner, Jan 06 2023

Status

approved