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#33 by Charles R Greathouse IV at Thu Sep 08 08:46:12 EDT 2022
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| PROG
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(MAGMAMagma) [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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Discussion
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| Thu Sep 08
| 08:46
| OEIS Server: https://oeis.org/edit/global/2944
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#32 by N. J. A. Sloane at Mon Apr 26 11:50:49 EDT 2021
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#31 by N. J. A. Sloane at Mon Apr 26 11:50:47 EDT 2021
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| NAME
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Resulting numberNumber of pieces after a sheet of paper is folded n times and cut diagonally.
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| COMMENTS
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For general n, foldFold 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)... altogethermaking n folds in all. Cut along the diagonal line from top left - to bottom right of the resulting small rectangle. CountSequence gives the number of pieces that are formed.
The even-numbered entries of this sequence are sequenceA343175 (essentially A085601. ). The odd numbered entries of thisare sequenceA343176 (essentially forA036562). [These n>2bisections are easy to analyze and have simpler formulas. - _N. J. A. Sloane_, Apr sequence26 A036562.2021]
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| CROSSREFS
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Cf. A036562, A085601, A343175, A343176.
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| STATUS
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approved
editing
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#30 by N. J. A. Sloane at Sun Apr 25 13:23:31 EDT 2021
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#29 by N. J. A. Sloane at Sun Apr 25 13:23:29 EDT 2021
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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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| STATUS
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approved
editing
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#28 by Alois P. Heinz at Wed Feb 05 11:52:43 EST 2020
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#27 by Stefano Spezia at Wed Feb 05 11:27:44 EST 2020
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#26 by Stefano Spezia at Wed Feb 05 11:27:37 EST 2020
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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
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| STATUS
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proposed
editing
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#25 by Colin Barker at Wed Feb 05 10:27:56 EST 2020
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#24 by Colin Barker at Wed Feb 05 10:27:34 EST 2020
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| LINKS
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Colin Barker, <a href="/A257418/b257418.txt">Table of n, a(n) for n = 0..1000</a>
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| FORMULA
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a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n>5. - Colin Barker, Feb 05 2020
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(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
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| STATUS
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approved
editing
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