%I #33 Sep 08 2022 08:46:12

%S 2,3,5,8,13,23,41,77,145,281,545,1073,2113,4193,8321,16577,33025,

%T 65921,131585,262913,525313,1050113,2099201,4197377,8392705,16783361,

%U 33562625,67121153,134234113,268460033,536903681,1073790977,2147549185,4295065601,8590065665

%N Number of pieces after a sheet of paper is folded n times and cut diagonally.

%C Fold a rectangular sheet of paper 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)... making n folds in all. Cut along the diagonal line from top left to bottom right of the resulting small rectangle. Sequence gives the number of pieces that are formed.

%C The even-numbered entries of this sequence are A343175 (essentially A085601). The odd numbered entries are A343176 (essentially A036562). [These bisections are easy to analyze and have simpler formulas. - _N. J. A. Sloane_, Apr 26 2021]

%H Colin Barker, <a href="/A257418/b257418.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-6,4).

%F a(n) = (2^n+2^(n/2)*(1+(-1)^n+3*sqrt(2)*(1-(-1)^n)/4)+2)/2 for n>1. (_Johan Nilsson_)

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

%F 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

%F a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n>5. - _Colin Barker_, Feb 05 2020

%F E.g.f.: (1/4)*(-2 - 2*x + 2*cosh(2*x) + 4*cosh(sqrt(2)*x) + 4*sinh(x) + 4*cosh(x)*(1 + sinh(x)) + 3*sqrt(2)*sinh(sqrt(2)*x)). - _Stefano Spezia_, Feb 05 2020

%e 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.

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

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

%t 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 *)

%o (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

%o (PARI) Vec((2 - 3*x - 4*x^2 + 5*x^3 - x^4 + 2*x^5) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)) + O(x^35)) \\ _Colin Barker_, Feb 05 2020

%o (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

%Y Cf. A036562, A085601, A343175, A343176.

%K nonn,easy

%O 0,1

%A _Dirk Frettlöh_, Apr 22 2015