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

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

Offset: 0

Author

S. Brunner, Jan 06 2023

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• 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.