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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
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.
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
Sequence in context: A171452 A049648 A211647 * A156039 A350477 A310767
KEYWORD
nonn,easy
AUTHOR
Harold M. Frost, III (halfrost(AT)charter.net), Dec 19 2007
EXTENSIONS
Edited, extended, and formulas by Nathaniel Johnston, Nov 11 2012
STATUS
approved