A133257 The number of edges on a piece of paper that has been folded n times (see comments for more precise definition).

4, 7, 11, 17, 25, 37, 53, 77, 109, 157, 221, 317, 445, 637, 893, 1277, 1789, 2557, 3581, 5117, 7165, 10237, 14333, 20477, 28669, 40957, 57341, 81917, 114685, 163837, 229373, 327677, 458749, 655357, 917501, 1310717, 1835005, 2621437, 3670013, 5242877, 7340029

Offset: 0

Author

Harold M. Frost, III (halfrost(AT)charter.net), Dec 19 2007

Keywords

• nonn
• easy

Comments

The angle each (straight) fold line makes with the long dimension of the original rectangle alternates in degrees as 90, 0, 90, 0, and so on. Each fold doubles the number of rectangles and halves the area of each rectangle.

The first five or six terms in the sequence can be verified experimentally with a standard piece of paper.

Examples

			When n = 0, the piece of paper hasn't been folded yet and has 4 edges. Thus a(0) = 4.
When n = 1, we have folded the piece of paper once. The fold splits 2 of the original edges in half, resulting in 6 edges, and it creates one new edge at the fold itself, for 7 edges in total. Thus a(1) = 7.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Crossrefs

Cf. A014577, A014707.

Programs

PARI

Vec(-(4*x^2-3*x-4)/((x-1)*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Jul 07 2014

Formula

a(n) = 2*a(n-2) + 3 for n >= 2.

a(n) = 2^((n-2)/2)*((7+5*sqrt(2))+(7-5*sqrt(2))*(-1)^n)-3.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3). G.f.: -(4*x^2-3*x-4) / ((x-1)*(2*x^2-1)). - Colin Barker, Jul 07 2014

Extensions

Edited, extended, and formulas by Nathaniel Johnston, Nov 11 2012