cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A085601 a(n) = 2 * (4^n + 2^n) + 1.

Original entry on oeis.org

5, 13, 41, 145, 545, 2113, 8321, 33025, 131585, 525313, 2099201, 8392705, 33562625, 134234113, 536903681, 2147549185, 8590065665, 34360000513, 137439477761, 549756862465, 2199025352705, 8796097216513, 35184380477441
Offset: 0

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jul 07 2003

Keywords

Comments

1. Begin with a square tile.
2. Place square tiles on each edge to form a diamond shape.
3. Count the tiles: a(0) = 5.
4. Add tiles to fill the enclosing square.
5. Go to step 2.

Links

Crossrefs

Cf. A028403, A046142, A056640, A064412.
Cf. A343175 (essentially the same).

Programs

  • Mathematica
    Table[2(4^n+2^n)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{5,13,41},30] (* Harvey P. Dale, Dec 30 2017 *)
  • PARI
    first(n) = Vec((5 - 22*x + 20*x^2)/(1 - 7*x + 14*x^2 - 8*x^3) + O(x^n)) \\ Iain Fox, Dec 30 2017

Formula

From R. J. Mathar, Apr 20 2009: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: -(5 - 22*x + 20*x^2)/((x - 1)*(2*x - 1)*(4*x - 1)).
(End)
E.g.f.: e^x + 2*(e^(2*x) + e^(4*x)). - Iain Fox, Dec 30 2017

Extensions

Edited by Franklin T. Adams-Watters and Don Reble, Aug 15 2006

A257418 Number of pieces after a sheet of paper is folded n times and cut diagonally.

Original entry on oeis.org

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
Offset: 0

Views

Author

Dirk Frettlöh, Apr 22 2015

Keywords

Comments

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.
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]

Examples

			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.

Links

Crossrefs

Cf. A036562, A085601, A343175, A343176.

Programs

  • 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
  • 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 *)
  • 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
  • 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

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
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n>5. - Colin Barker, Feb 05 2020
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

A343176 a(0)=3; for n > 0, a(n) = 2^(2*n) + 3*2^(n-1) + 1.

Original entry on oeis.org

3, 8, 23, 77, 281, 1073, 4193, 16577, 65921, 262913, 1050113, 4197377, 16783361, 67121153, 268460033, 1073790977, 4295065601, 17180065793, 68719869953, 274878693377, 1099513200641, 4398049656833, 17592192335873, 70368756760577, 281475001876481, 1125899957174273, 4503599728033793
Offset: 0

Views

Author

N. J. A. Sloane, Apr 26 2021

Keywords

Comments

A bisection of A257418. Apart from first term, same as A036562.

Links

Crossrefs

Cf. A257418, A036562, A343175.

Formula

From Chai Wah Wu, Apr 26 2021: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n > 3.
G.f.: -(4*x - 3)*(x^2 + 3*x - 1)/((x - 1)*(2*x - 1)*(4*x - 1)). (End)

Extensions

a(17)-a(26) from Martin Ehrenstein, Apr 26 2021
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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