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A257418 Resulting number of pieces after a sheet of paper is folded n times and cut diagonally. 0
2, 3, 5, 8, 13, 23, 41, 77, 145, 281, 545, 1073, 2113, 4193, 8321, 16577, 33025, 65921, 131585, 262913, 525313, 1050113, 2099201, 4197377, 8392705, 16783361, 33562625, 67121153, 134234113, 268460033, 536903681, 1073790977, 2147549185, 4295065601, 8590065665 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

For general n, fold a rectangular sheet of paper (A4, say) in half (fold lower half up), and again into half (left half to the right), and again (lower half up), and again (left half to the right)... altogether n folds. Cut along the diagonal top left - bottom right of the resulting small rectangle. Count the pieces.

The even-numbered entries of this sequence are sequence A085601. The odd numbered entries of this sequence for n>2 are sequence A036562.

LINKS

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

FORMULA

a(n) = (2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2 for n>1. (Johan Nilsson)

a(0) = 2, a(1) = 3, a(n+1) = 2*a(n)-2^(floor((n-1)/2))-1.

G.f.: -(2*x^5-x^4+5*x^3-4*x^2-3*x+2)/((x-1)*(2*x-1)*(2*x^2-1)). - Alois P. Heinz, Apr 23 2015

EXAMPLE

n=1: Take a rectangular sheet of paper and fold it in half. Cutting along the diagonal of the resulting rectangle yields 3 smaller pieces of paper.

n=0: Cutting the sheet of paper (without any folding) along the diagonal yields two pieces.

MAPLE

2, seq(floor((2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2), n=1..25);

MATHEMATICA

Table[Floor[(2^n + 2^(n/2)*(1 + (-1)^n + 3 Sqrt[2]*(1 - (-1)^n)/4) + 2)/2], {n, 0, 25}] (* Michael De Vlieger, Apr 24 2015 *)

PROG

(PARI) concat(2, vector(30, n, round((2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2))) \\ Derek Orr, Apr 27 2015

(MAGMA) [2, 3, 5, 8] cat [Floor((2^n+2^(n/2)*(1+(-1)^n+3*Sqrt(2)*(1-(-1)^n)/4)+2)/2):n in [4..40]]; // Vincenzo Librandi, May 05 2015

CROSSREFS

Cf. A036562, A085601.

Sequence in context: A018067 A068202 A096796 * A125730 A074030 A024318

Adjacent sequences:  A257415 A257416 A257417 * A257419 A257420 A257421

KEYWORD

nonn,easy

AUTHOR

Dirk Frettlöh, Apr 22 2015

STATUS

approved

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Last modified January 17 09:54 EST 2020. Contains 330949 sequences. (Running on oeis4.)