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

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(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

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For general n=1: Take , fold a rectangular sheet of paper (A4, say) and fold it in half. Cutting along the diagonal of the resulting rectangle yields 3 smaller pieces of paper. General n: Fold the 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)... altogether n folds. Cut along the diagonal top left - bottom right of the resulting small rectangle. Count the pieces.

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.

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Number Resulting number of pieces of after a piece sheet of paper is folded n times and cut diagonally.

n=1: Take a piece rectangular sheet of paper (A4, say) and fold it in half. Cutting along the diagonal of the resulting rectangle yields 3 smaller pieces of paper. General n: Fold the piece 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)... 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.

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

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