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A133257 The number of edges on a piece of paper that has been folded n times (see comments for more precise definition). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

LINKS

Table of n, a(n) for n=0..40.

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

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

EXAMPLE

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.

PROG

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

CROSSREFS

Cf. A014577, A014707.

Sequence in context: A171452 A049648 A211647 * A156039 A310767 A207871

Adjacent sequences:  A133254 A133255 A133256 * A133258 A133259 A133260

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified March 23 04:46 EDT 2019. Contains 321422 sequences. (Running on oeis4.)